Mister Exam

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Sum of series:
1/((3n-2)*(3n+1))
n*3^n+1
factorial(n)^2/factorial(k^n)
arctg(x)
Identical expressions
(x+ four)^n/n^ two
(x plus 4) to the power of n divide by n squared
(x plus four) to the power of n divide by n to the power of two
(x+4)n/n2
x+4n/n2
(x+4)^n/n²
(x+4) to the power of n/n to the power of 2
x+4^n/n^2
(x+4)^n divide by n^2
Similar expressions
(x-4)^n/n^2

Sum of series (x+4)^n/n^2

The solution

You have entered [src]

  oo          
____          
\   `         
 \           n
  \   (x + 4) 
   )  --------
  /       2   
 /       n    
/___,         
n = 1

\sum_{n=1}^{\infty} \frac{\left(x + 4\right)^{n}}{n^{2}}

Sum((x + 4)^n/n^2, (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{\left(x + 4\right)^{n}}{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} = \frac{1}{n^{2}}

and

x_{0} = -4

d = 1

c = 1

then

R = -4 + \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n^{2}}\right)

Let's take the limit
we find

R = -3

The answer [src]

/polylog(2, 4 + x)  for |4 + x| <= 1
|                                   
|   oo                              
| ____                              
| \   `                             
|  \           n                    
<   \   (4 + x)                     
|    )  --------       otherwise    
|   /       2                       
|  /       n                        
| /___,                             
| n = 1                             
\

\begin{cases} \operatorname{Li}_{2}\left(x + 4\right) & \text{for}\: \left|{x + 4}\right| \leq 1 \\\sum_{n=1}^{\infty} \frac{\left(x + 4\right)^{n}}{n^{2}} & \text{otherwise} \end{cases}

Piecewise((polylog(2, 4 + x), |4 + x| <= 1), (Sum((4 + x)^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
```

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Identical expressions

Similar expressions

Sum of series (x+4)^n/n^2

The solution

Examples of finding the sum of a series