Mathematics: Find the intervals of convergence for the function. f(x)=sum_(n=1)^infinity x**n/n**2 32324243232 incorrect: your answer is incorrect. (b) find the intervals of convergence for f .

Find the intervals of convergence for the function.

Ответ:

mastershadow2018

13.03.2022

$\displaystyle\sum_{n\ge1}\frac{x^n}{n^2}$

will converge as long as

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{(n+1)^2}}{\frac{x^n}{n^2}}\right|$

according to the ratio test. We have

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{(n+1)^2}}{\frac{x^n}{n^2}}\right|=|x|\lim_{n\to\infty}\frac{n^2}{(n+1)^2}=|x|\left(\lim_{n\to\infty}\frac n{n+1}\right)^2=|x|$

so the sum will certainly converge whenever

, but we also know the series will converge at the endpoints. $\displaystyle\sum\frac1{n^2}$ is a convergent

-series, which means $\displaystyle\sum\frac{(-1)^n}{n^2}$ is also (absolutely) convergent, so in fact the interval of convergence is $-1\le x\le1$ (endpoints included).

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Ответ:

noah12345678

10.05.2023

Mathematics

Any line can be graphed using two points. Select two x x values, and plug them into the equation to find the corresponding y y values.

Step-by-step explanation:

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Find the intervals of convergence for the function. f(x)=sum_(n=1)^infinity xn/n2 32324243232 incorrect: your answer is incorrect. (b) find the intervals of convergence for f '.

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Find the intervals of convergence for the function. f(x)=sum_(n=1)^infinity x**n/n**2 32324243232 incorrect: your answer is incorrect. (b) find the intervals of convergence for f '.

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Find the intervals of convergence for the function. f(x)=sum_(n=1)^infinity xn/n2 32324243232 incorrect: your answer is incorrect. (b) find the intervals of convergence for f '.