Mister Exam

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Sum of series:
n^2/2^n
ln^n(x)/n!
ln(n/10)/(exp(n/10)-1)
ln(n/10)/(exp(n/10)+1)
Identical expressions
ln(n/ ten)/(exp(n/ ten)- one)
ln(n divide by 10) divide by ( exponent of (n divide by 10) minus 1)
ln(n divide by ten) divide by ( exponent of (n divide by ten) minus one)
lnn/10/expn/10-1
ln(n divide by 10) divide by (exp(n divide by 10)-1)
Similar expressions
ln(n/10)/(exp(n/10)+1)

Sum of series/ ln(n/10)/(exp(n/10)-1)

Sum of series ln(n/10)/(exp(n/10)-1)

The solution

You have entered [src]

   oo          
______         
\     `        
 \         /n \
  \     log|--|
   \       \10/
    \   -------
    /    n     
   /     --    
  /      10    
 /      e   - 1
/_____,        
 n = 1

\sum_{n=1}^{\infty} \frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} - 1}

Sum(log(n/10)/(exp(n/10) - 1), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} - 1}

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{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} - 1}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty} \left|{\frac{\left(e^{\frac{n}{10} + \frac{1}{10}} - 1\right) \log{\left(\frac{n}{10} \right)}}{\left(e^{\frac{n}{10}} - 1\right) \log{\left(\frac{n}{10} + \frac{1}{10} \right)}}}\right|

Let's take the limit
we find

False

False

False

The rate of convergence of the power series

Numerical answer [src]

-34.2737296704091840084709480062

-34.2737296704091840084709480062

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 ln(n/10)/(exp(n/10)-1)

The solution

Examples of finding the sum of a series