Mister Exam

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Sum of series:
n^2*sin(5/(3^n))
n*2^n
n^(1/n)
(n^2+5)/2^n
Identical expressions
log(n/(n+ one))
logarithm of (n divide by (n plus 1))
logarithm of (n divide by (n plus one))
logn/n+1
log(n divide by (n+1))
Similar expressions
log(n/(n-1))

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

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

The solution

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

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

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

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

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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Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series