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Sum of series:
1/((3n-2)*(3n+1))
(n+2/2n)^n2
n*3^n+1
ln(k/(k+1))
Identical expressions
(n+ two /2n)^n2
(n plus 2 divide by 2n) to the power of n2
(n plus two divide by 2n) to the power of n2
(n+2/2n)n2
n+2/2nn2
n+2/2n^n2
(n+2 divide by 2n)^n2
Similar expressions
(n-2/2n)^n2

Sum of series (n+2/2n)^n2

The solution

You have entered [src]

  oo           
 ___           
 \  `          
  \          n2
  /   (n + n)  
 /__,          
n = 1

\sum_{n=1}^{\infty} \left(n + n\right)^{n_{2}}

Sum((n + n)^n2, (n, 1, oo))

The radius of convergence of the power series

Given number:

\left(n + n\right)^{n_{2}}

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} = \left(2 n\right)^{n_{2}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\left(2 n\right)^{\operatorname{re}{\left(n_{2}\right)}} \left(2 n + 2\right)^{- \operatorname{re}{\left(n_{2}\right)}}\right)

Let's take the limit
we find

True

False

The answer [src]

/ n2                       
|2  *zeta(-n2)  for n2 < -1
|                          
|  oo                      
| ___                      
< \  `                     
|  \        n2             
|  /   (2*n)     otherwise 
| /__,                     
|n = 1                     
\

\begin{cases} 2^{n_{2}} \zeta\left(- n_{2}\right) & \text{for}\: n_{2} < -1 \\\sum_{n=1}^{\infty} \left(2 n\right)^{n_{2}} & \text{otherwise} \end{cases}

Piecewise((2^n2*zeta(-n2), n2 < -1), (Sum((2*n)^n2, (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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Identical expressions

Similar expressions

Sum of series (n+2/2n)^n2

The solution

Examples of finding the sum of a series