Sum of series a^n

The solution

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  oo    
 ___    
 \  `   
  \    n
  /   a 
 /__,   
n = 1

$$\sum_{n=1}^{\infty} a^{n}$$

Sum(a^n, (n, 1, oo))

The radius of convergence of the power series

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

The answer [src]

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

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

Piecewise((a/(1 - a), |a| < 1), (Sum(a^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
```

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Sum of series a^n

The solution

Examples of finding the sum of a series