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Sum of series/ 2^n/n^2

Sum of series 2^n/n^2

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 \     n
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  /    2
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n = 1

\sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}}

Sum(2^n/n^2, (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{2^{n}}{n^{2}}

It is a series of species

a_{n} \left(c x - x_{0}\right)^{d n}

- power series.
The radius of convergence of a power series can be calculated by the formula:

R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}

In this case

a_{n} = \frac{1}{n^{2}}

and

x_{0} = -2

d = 1

c = 0

then

R = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n^{2}}\right)\right)

False

The rate of convergence of the power series

The answer [src]

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\infty

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Numerical answer

The series diverges

The graph