Mister Exam

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Sum of series:
1/(2*n-1)*(2*n+1)
sqrt(n)*(x+3)^n/n^2+1
sin(n*x)/n^2
ln((n(n+2))/(n+1)^2)
Identical expressions
one /ln(five ^n)
1 divide by ln(5 to the power of n)
one divide by ln(five to the power of n)
1/ln(5n)
1/ln5n
1/ln5^n
1 divide by ln(5^n)

Sum of series/ 1/ln(5^n)

Sum of series 1/ln(5^n)

The solution

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  oo         
____         
\   `        
 \       1   
  \   -------
  /      / n\
 /    log\5 /
/___,        
n = 1

$$\sum_{n=1}^{\infty} \frac{1}{\log{\left(5^{n} \right)}}$$

Sum(1/log(5^n), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{1}{\log{\left(5^{n} \right)}}$$
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
$$a_{n} = \frac{1}{\log{\left(5^{n} \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\log{\left(5^{n + 1} \right)}}{\log{\left(5^{n} \right)}}}\right|$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

oo

$$\infty$$

oo

Numerical answer

The series diverges

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

Sum of series 1/ln(5^n)

The solution

Examples of finding the sum of a series