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Sum of series n^2*x^n
The solution
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oo ___ \ ` \ 2 n / n *x /__, n = 1
$$\sum_{n=1}^{\infty} n^{2} x^{n}$$
Sum(n^2*x^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n^{2} x^{n}$$
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
$$a_{n} = n^{2}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{n^{2}}{\left(n + 1\right)^{2}}\right)$$
Let's take the limit
we find
$$R = 1$$
$$n^{2} x^{n}$$
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
$$a_{n} = n^{2}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{n^{2}}{\left(n + 1\right)^{2}}\right)$$
Let's take the limit
we find
$$R = 1$$
The answer
[src]
/x*(-1 - x) |---------- for |x| < 1 | 3 |(-1 + x) | | oo < ___ | \ ` | \ 2 n | / n *x otherwise | /__, |n = 1 \
$$\begin{cases} \frac{x \left(- x - 1\right)}{\left(x - 1\right)^{3}} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=1}^{\infty} n^{2} x^{n} & \text{otherwise} \end{cases}$$
Piecewise((x*(-1 - x)/(-1 + x)^3, |x| < 1), (Sum(n^2*x^n, (n, 1, oo)), True))
Examples of finding the sum of a series
- The Sum of the Power Series
x^n/n
(x-1)^n
- Factorial
1/2^(n!)
n^2/n!
x^n/n!
k!/(n!*(n+k)!)
- Flint Hills Series
csc(n)^2/n^3
- Basel problem
1/n^2
1/n^4
1/n^6
- Harmonic series
1/n
- Grandi's series
(-1)^n
- Alternating series
(-1)^(n + 1)/n
(n + 2)*(-1)^(n - 1)
(3*n - 1)/(-5)^n
(-1)^(n - 1)*n/(6*n - 5)
- Newton–Mercator series
(-1)^(n + 1)/n*x^n
- Exam the series for convergence
(3*n - 1)/(-5)^n