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Sum of series/ n*3^n+1

Sum of series n*3^n+1

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  \   /   n    \
  /   \n*3  + 1/
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n = 1

\sum_{n=1}^{\infty} \left(3^{n} n + 1\right)

Sum(n*3^n + 1, (n, 1, oo))

The radius of convergence of the power series

Given number:

3^{n} n + 1

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} = 3^{n} n + 1

and

x_{0} = 0

d = 0

c = 1

then

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

False

False

False

The rate of convergence of the power series

The answer [src]

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\infty

oo

Numerical answer

The series diverges

The graph