Mister Exam

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Sum of series:
x^n/sqrt(n+1)
x^2j
3/n
1-cos(pi/n)
Identical expressions
(x^n)/(n*(n+ one))
(x to the power of n) divide by (n multiply by (n plus 1))
(x to the power of n) divide by (n multiply by (n plus one))
(xn)/(n*(n+1))
xn/n*n+1
(x^n)/(n(n+1))
(xn)/(n(n+1))
xn/nn+1
x^n/nn+1
(x^n) divide by (n*(n+1))
Similar expressions
(x^n)/(n*(n-1))

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

The solution

You have entered [src]

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

\sum_{n=1}^{\infty} \frac{x^{n}}{n \left(n + 1\right)}

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

The radius of convergence of the power series

Given number:

\frac{x^{n}}{n \left(n + 1\right)}

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{1}{n \left(n + 1\right)}

and

x_{0} = 0

d = 1

c = 1

then

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

Let's take the limit
we find

R = 1

The answer [src]

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

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

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

How to use it?

Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series