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Sum of series/ n^2/3^n

Sum of series n^2/3^n

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  oo    
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\   `   
 \     2
  \   n 
   )  --
  /    n
 /    3 
/___,   
n = 1

\sum_{n=1}^{\infty} \frac{n^{2}}{3^{n}}

Sum(n^2/3^n, (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{n^{2}}{3^{n}}

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} = n^{2}

and

x_{0} = -3

d = -1

c = 0

then

\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(\frac{n^{2}}{\left(n + 1\right)^{2}}\right)\right)

False

R = 0

The rate of convergence of the power series

The answer [src]

3/2

\frac{3}{2}

3/2

Numerical answer [src]

1.50000000000000000000000000000

1.50000000000000000000000000000

The graph