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Sum of series:
sqrt(n)*sin(pi/n^2)
-6n(6n+6)+(6n-6)(6+6n)
(n*x^n)/(n+1)
1/(n(n+1)(n+2))
Identical expressions
(n*x^n)/(n+ one)
(n multiply by x to the power of n) divide by (n plus 1)
(n multiply by x to the power of n) divide by (n plus one)
(n*xn)/(n+1)
n*xn/n+1
(nx^n)/(n+1)
(nxn)/(n+1)
nxn/n+1
nx^n/n+1
(n*x^n) divide by (n+1)
Similar expressions
(n*x^n)/(n-1)

Sum of series (n*x^n)/(n+1)

The solution

You have entered [src]

  oo       
____       
\   `      
 \        n
  \    n*x 
  /   -----
 /    n + 1
/___,      
n = 0

\sum_{n=0}^{\infty} \frac{n x^{n}}{n + 1}

Sum((n*x^n)/(n + 1), (n, 0, oo))

The radius of convergence of the power series

Given number:

\frac{n x^{n}}{n + 1}

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} = \frac{n}{n + 1}

and

x_{0} = 0

d = 1

c = 1

then

R = \lim_{n \to \infty}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right)

Let's take the limit
we find

R = 1

The answer [src]

/  /    2      2*log(1 - x)\             
|x*|- ------ + ------------|             
|  |   2             2     |             
|  \  x  - x        x      /             
|---------------------------  for |x| < 1
|             2                          
|                                        
|          oo                            
<        ____                            
|        \   `                           
|         \        n                     
|          \    n*x                      
|          /   -----           otherwise 
|         /    1 + n                     
|        /___,                           
|        n = 0                           
\

\begin{cases} \frac{x \left(- \frac{2}{x^{2} - x} + \frac{2 \log{\left(1 - x \right)}}{x^{2}}\right)}{2} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=0}^{\infty} \frac{n x^{n}}{n + 1} & \text{otherwise} \end{cases}

Piecewise((x*(-2/(x^2 - x) + 2*log(1 - x)/x^2)/2, |x| < 1), (Sum(n*x^n/(1 + 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
```

Other calculators

How to use it?

Identical expressions

Similar expressions

Sum of series (n*x^n)/(n+1)

The solution

Examples of finding the sum of a series