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n/3^n

Sum of series n/3^n



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The solution

You have entered [src]
  oo    
____    
\   `   
 \    n 
  \   --
  /    n
 /    3 
/___,   
n = 1   
$$\sum_{n=1}^{\infty} \frac{n}{3^{n}}$$
Sum(n/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n}{3^{n}}$$
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
$$a_{n} = n$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(\frac{n}{n + 1}\right)\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src]
3/4
$$\frac{3}{4}$$
3/4
Numerical answer [src]
0.750000000000000000000000000000
0.750000000000000000000000000000
The graph
Sum of series n/3^n

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