Mister Exam

Sum of series nx^n

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The solution

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  oo
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\
\      n
/   n*x
/__,
n = 1     
$$\sum_{n=1}^{\infty} n x^{n}$$
Sum(n*x^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n x^{n}$$
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
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{n}{n + 1}\right)$$
Let's take the limit
we find
$$R = 1$$
/    x
| --------   for |x| < 1
|        2
| (1 - x)
|
|  oo
< ___
| \  
|  \      n
|  /   n*x    otherwise
| /__,
|n = 1
\                       
$$\begin{cases} \frac{x}{\left(1 - x\right)^{2}} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=1}^{\infty} n x^{n} & \text{otherwise} \end{cases}$$
Piecewise((x/(1 - x)^2, |x| < 1), (Sum(n*x^n, (n, 1, oo)), True))