Sum of series a*x^n

The solution

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  oo      
 ___      
 \  `     
  \      n
  /   a*x 
 /__,     
n = 0

$$\sum_{n=0}^{\infty} a x^{n}$$

Sum(a*x^n, (n, 0, oo))

The radius of convergence of the power series

Given number:
$$a x^{n}$$
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
$$a_{n} = a$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty} 1$$
Let's take the limit
we find
$$R = 1$$

The answer [src]

  //   1                 \
  || -----    for |x| < 1|
  || 1 - x               |
  ||                     |
  ||  oo                 |
a*|< ___                 |
  || \  `                |
  ||  \    n             |
  ||  /   x    otherwise |
  || /__,                |
  \\n = 0                /

$$a \left(\begin{cases} \frac{1}{1 - x} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=0}^{\infty} x^{n} & \text{otherwise} \end{cases}\right)$$

a*Piecewise((1/(1 - x), |x| < 1), (Sum(x^n, (n, 0, 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
```

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Identical expressions

Sum of series a*x^n

The solution

Examples of finding the sum of a series