Раздел I. Введение в анализ — § 2. Теория последовательностей

Демидович — задача 57

Б. П. Демидович, «Сборник задач и упражнений по математическому анализу». Условие и подробное решение по шагам.

Предполагая, что nn пробегает натуральный ряд чисел, определить значение следующего выражения:

Задача 57

lim⁡n→∞(2 24 28⋯22n).\lim_{n\to\infty}\left(\sqrt2\,\sqrt[4]2\,\sqrt[8]2\cdots\sqrt[2^n]2\right).

Опирается на задачу 7

Решение

Идея

Все множители являются степенями одного и того же числа 22, поэтому
при умножении достаточно сложить их показатели. Эта сумма геометрическая и
вычисляется точно. После свёртывания произведения останется обосновать, что
212n→12^{\frac1{2^n}}\to1; сделаем это непосредственно с помощью неравенства
Бернулли из задачи 7, не ссылаясь на ещё не доказанную непрерывность
показательной функции.
Обозначим произведение через PnP_n. Тогда

Pn=212214⋯212n=2∑k=1n2−k(поформулегеометрическойпрогрессии)=21−2−n=2212n.P_n =2^{\frac12}2^{\frac14}\cdots2^{\frac1{2^n}} =2^{\sum_{k=1}^{n}2^{-k}}
(по формуле геометрической прогрессии) =2^{1-2^{-n}}
=\frac{2}{2^{\frac1{2^n}}}.
Положимun=212n−1>0.u_n=2^{\frac1{2^n}}-1>0.Тогда(1+un)2n=2.(1+u_n)^{2^n}=2.По строгому неравенству Бернулли из задачи 7
Напоминание неравенства Бернулли: (1+x)n>=1+xn при x>−1,n>1иравенстводостигаетсятолькопри(1+x)^n >= 1 + xn \text{ при } x > -1, n > 1 и равенство достигается только приx = 0$<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>u</mi><mi>n</mi></msub><msup><mo stretchy="false">)</mo><msup><mn>2</mn><mi>n</mi></msup></msup><mo>&gt;</mo><mn>1</mn><mo>+</mo><msup><mn>2</mn><mi>n</mi></msup><msub><mi>u</mi><mi>n</mi></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">(1+u_n)^{2^n}&gt;1+2^n u_n.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.18em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8644em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span>Следовательно,<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><mo>&gt;</mo><mn>1</mn><mo>+</mo><msup><mn>2</mn><mi>n</mi></msup><msub><mi>u</mi><mi>n</mi></msub><mo separator="true">,</mo><mspace width="2em"/><mn>0</mn><mo>&lt;</mo><msub><mi>u</mi><mi>n</mi></msub><mo>&lt;</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">2&gt;1+2^n u_n,
\qquad
0&lt;u_n&lt;\frac1{2^n}.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9088em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5904em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">.</span></span></span></span></span>По теореме о зажатой последовательностиu_n\to0,тоесть<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msup><mn>2</mn><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></msup><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>u</mi><mi>n</mi></msub><mo>⟶</mo><mn>1.</mn></mrow><annotationencoding="application/x−tex">212n=1+un⟶1.</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1.004em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:1.004em;"><spanstyle="top:−3.413em;margin−right:0.05em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mopennulldelimitersizingreset−size3size6"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8443em;"><spanstyle="top:−2.656em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmtight">2</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7246em;"><spanstyle="top:−2.794em;margin−right:0.1em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="mordmathnormalmtight">n</span></span></span></span></span></span></span></span></span></span><spanstyle="top:−3.2255em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−linemtight"style="border−bottom−width:0.049em;"></span></span><spanstyle="top:−3.384em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmtight">1</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.344em;"><span></span></span></span></span></span><spanclass="mclosenulldelimitersizingreset−size3size6"></span></span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">1</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.661em;vertical−align:−0.15em;"></span><spanclass="mord"><spanclass="mordmathnormal">u</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1514em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">⟶</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1.</span></span></span></span></span>Поэтому<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msub><mi>P</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>2</mn><msup><mn>2</mn><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></msup></mfrac><mo>⟶</mo><mn>2.</mn></mrow><annotationencoding="application/x−tex">Pn=2212n⟶2.</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8333em;vertical−align:−0.15em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.13889em;">P</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1514em;"><spanstyle="top:−2.55em;margin−left:−0.1389em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.151em;vertical−align:−0.8296em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.1704em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"><spanclass="mord">2</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.9396em;"><spanstyle="top:−3.3486em;margin−right:0.05em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mopennulldelimitersizingreset−size3size6"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8443em;"><spanstyle="top:−2.656em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmtight">2</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7246em;"><spanstyle="top:−2.794em;margin−right:0.1em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="mordmathnormalmtight">n</span></span></span></span></span></span></span></span></span></span><spanstyle="top:−3.2255em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−linemtight"style="border−bottom−width:0.049em;"></span></span><spanstyle="top:−3.384em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmtight">1</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.344em;"><span></span></span></span></span></span><spanclass="mclosenulldelimitersizingreset−size3size6"></span></span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8296em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">⟶</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">2.</span></span></span></span></span>Последнийкореньвпроизведенииимеетпоказатель, то есть<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mn>2</mn><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></msup><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>u</mi><mi>n</mi></msub><mo>⟶</mo><mn>1.</mn></mrow><annotation encoding="application/x-tex">2^{\frac1{2^n}}=1+u_n\longrightarrow1.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.004em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.004em;"><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7246em;"><span style="top:-2.794em;margin-right:0.1em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.661em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟶</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1.</span></span></span></span></span>Поэтому<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>2</mn><msup><mn>2</mn><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></msup></mfrac><mo>⟶</mo><mn>2.</mn></mrow><annotation encoding="application/x-tex">P_n=\frac2{2^{\frac1{2^n}}}\longrightarrow2.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.151em;vertical-align:-0.8296em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.1704em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7246em;"><span style="top:-2.794em;margin-right:0.1em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8296em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟶</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2.</span></span></span></span></span>Последний корень в произведении имеет показатель\frac1{2^n},поэтомусуммапоказателейдействительноравна,
поэтому сумма показателей действительно равна1-2^{-n}.Всеоснованияположительны,аоценка. Все основания
положительны, а оценка0непрерывности показательной функции; преобразования степеней и последующий
предельный переход поэтому корректны.

Проверка

2.2.
Демидович № 57 — решение