Answer to Question #103940 in Calculus for BIVEK SAH

Question #103940
Test the conditional convergence of the series


1-1/root2+1/root3-1/root4+....
1
Expert's answer
2020-03-10T10:38:41-0400

112+1314+...+(1)n1n+...()an=(1)n1n=(1)n1n12.1-\frac{1}{\sqrt2}+\frac{1}{\sqrt3}-\frac{1}{\sqrt4}+...+\frac{(-1)^{n-1}}{\sqrt n}+...(*)\\ a_n=\frac{(-1)^{n-1}}{\sqrt n}=\frac{(-1)^{n-1}}{ n^\frac{1}{2}}.

Consider a series of modules

1+12+13+14+...+1n+...()1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+\frac{1}{\sqrt n}+...(**)\\

Series

n=11nα\sum\limits_{n=1}^{\infty} \frac{1}{n^\alpha}

convergent if α>1\alpha>1 , divergent if α1\alpha \leq1 .

In our case α=12<1\alpha=\frac{1}{2}<1 then series (**) is divergent.

We examine the series (*) on the basis of Leibniz

1)a1>a2>...>an>...1) |a_1|>|a_2|>...>|a_n|>...

in our case

1>12>13>...>1n>...2)limnan=limn1n=0.1>\frac{1}{\sqrt 2}>\frac{1}{\sqrt 3}>...>\frac{1}{\sqrt n}>...\\ 2) \lim\limits_{n\mapsto \infty}|a_n|=\lim\limits_{n\mapsto \infty} \frac{1}{\sqrt n}=0.

According to Leibniz, the series (*) is convergent.

 Since the series (**) is divergent and the series (*) is convergent, the series (*) is conventionally convergent.


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