Mister Exam

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Sum of series:
15/((7-6n)*(1-6n))
1/n^(1+1/n)
exp(-2)*x^n*2^n/factorial(n)
arcsin^n(2/(n+1))
Identical expressions
one /n^(one + one /n)
1 divide by n to the power of (1 plus 1 divide by n)
one divide by n to the power of (one plus one divide by n)
1/n(1+1/n)
1/n1+1/n
1/n^1+1/n
1 divide by n^(1+1 divide by n)
Similar expressions
1/n^(1-1/n)

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

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

The solution

You have entered [src]

  oo         
_____        
\    `       
 \       1   
  \    ------
   \        1
   /    1 + -
  /         n
 /     n     
/____,       
n = 1

$$\sum_{n=1}^{\infty} \frac{1}{n^{1 + \frac{1}{n}}}$$

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

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

The answer [src]

  oo         
____         
\   `        
 \          1
  \    -1 - -
  /         n
 /    n      
/___,        
n = 1

$$\sum_{n=1}^{\infty} n^{-1 - \frac{1}{n}}$$

Sum(n^(-1 - 1/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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Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series