Mister Exam

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Sum of series:
1+1/factorial(n)
(n+1)^n/(n^(n+2))
cos(n)/n
(1/sqrt(n))-(1/sqrt(n+1))
Identical expressions
(n+ one)^n/(n^(n+ two))
(n plus 1) to the power of n divide by (n to the power of (n plus 2))
(n plus one) to the power of n divide by (n to the power of (n plus two))
(n+1)n/(n(n+2))
n+1n/nn+2
n+1^n/n^n+2
(n+1)^n divide by (n^(n+2))
Similar expressions
(n+1)^n/(n^(n-2))
(n-1)^n/(n^(n+2))

Sum of series/ (n+1)^n/(n^(n+2))

Sum of series (n+1)^n/(n^(n+2))

The solution

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

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

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

The radius of convergence of the power series

Given number:
$$\frac{\left(n + 1\right)^{n}}{n^{n + 2}}$$
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
$$a_{n} = n^{- n - 2} \left(n + 1\right)^{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(n^{- n - 2} \left(n + 1\right)^{n} \left(n + 1\right)^{n + 3} \left(n + 2\right)^{- n - 1}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

  oo                  
 ___                  
 \  `                 
  \    -2 - n        n
  /   n      *(1 + n) 
 /__,                 
n = 1

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

Sum(n^(-2 - n)*(1 + n)^n, (n, 1, oo))

The graph

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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How to use it?

Identical expressions

Similar expressions

Sum of series (n+1)^n/(n^(n+2))

The solution

Examples of finding the sum of a series