Sum of series x^(n)

The solution

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 \  `   
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n = 1

\sum_{n=1}^{\infty} x^{n}

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

The radius of convergence of the power series

Given number:

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} = 1

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]

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

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

Piecewise((x/(1 - x), |x| < 1), (Sum(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
```

Other calculators

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Identical expressions

Sum of series x^(n)

The solution

Examples of finding the sum of a series