Mister Exam

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Sum of series:
1/n^6
ln((n+2)/n)
(3^n+9^n)/13^n
((-1)^(n+1))/(ln(n+1))
Identical expressions
((- one)^(n+ one))/(ln(n+ one))
(( minus 1) to the power of (n plus 1)) divide by (ln(n plus 1))
(( minus one) to the power of (n plus one)) divide by (ln(n plus one))
((-1)(n+1))/(ln(n+1))
-1n+1/lnn+1
-1^n+1/lnn+1
((-1)^(n+1)) divide by (ln(n+1))
Similar expressions
((-1)^(n+1))/(ln(n-1))
((1)^(n+1))/(ln(n+1))
((-1)^(n-1))/(ln(n+1))

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

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

The solution

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

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

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

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

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 ((-1)^(n+1))/(ln(n+1))

The solution

Examples of finding the sum of a series