Mister Exam

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Sum of series:
tg^(2)*(6/(2n)^(1/3))*(x-24)^n
1/(2*n-1)*(2*n+1)
cos(kx)
((-1)^(n-1))*(5^n+2)*x^n
Identical expressions
(- one)^n*ln(n/(n+ one))
( minus 1) to the power of n multiply by ln(n divide by (n plus 1))
( minus one) to the power of n multiply by ln(n divide by (n plus one))
(-1)n*ln(n/(n+1))
-1n*lnn/n+1
(-1)^nln(n/(n+1))
(-1)nln(n/(n+1))
-1nlnn/n+1
-1^nlnn/n+1
(-1)^n*ln(n divide by (n+1))
Similar expressions
(-1)^n*ln(n/(n-1))
(1)^n*ln(n/(n+1))

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

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

The solution

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

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

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

The radius of convergence of the power series

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

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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Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series