Mister Exam

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Sum of series:
((-1)^(n+1))/(2^n)
(-1)^n*x^n/(2*n+1)
(x^n)/(n*(n+1))
k*x+b
Identical expressions
(three -sinn)/(n-lnn)
(3 minus sinus of n) divide by (n minus lnn)
(three minus sinus of n) divide by (n minus lnn)
3-sinn/n-lnn
(3-sinn) divide by (n-lnn)
Similar expressions
(3-sinn)/(n+lnn)
(3+sinn)/(n-lnn)

Sum of series/ (3-sinn)/(n-lnn)

Sum of series (3-sinn)/(n-lnn)

The solution

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  oo            
 ___            
 \  `           
  \   3 - sin(n)
   )  ----------
  /   n - log(n)
 /__,           
n = 1

$$\sum_{n=1}^{\infty} \frac{3 - \sin{\left(n \right)}}{n - \log{\left(n \right)}}$$

Sum((3 - sin(n))/(n - log(n)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{3 - \sin{\left(n \right)}}{n - \log{\left(n \right)}}$$
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
$$a_{n} = \frac{3 - \sin{\left(n \right)}}{n - \log{\left(n \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(\sin{\left(n \right)} - 3\right) \left(n - \log{\left(n + 1 \right)} + 1\right)}{\left(n - \log{\left(n \right)}\right) \left(\sin{\left(n + 1 \right)} - 3\right)}}\right|$$
Let's take the limit
we find
$$1 = \left|{\left\langle -2, - \frac{1}{2}\right\rangle}\right|$$

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 (3-sinn)/(n-lnn)

The solution

Examples of finding the sum of a series