Mister Exam

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Sum of series:
38
arctan(n)/(n!)^2
arctg(5/n)/n!
ln((n^2+n+3)/(n^2+n+2))
Identical expressions
(one /n)*sin(x/n)
(1 divide by n) multiply by sinus of (x divide by n)
(one divide by n) multiply by sinus of (x divide by n)
(1/n)sin(x/n)
1/nsinx/n
(1 divide by n)*sin(x divide by n)

Sum of series (1/n)*sin(x/n)

The solution

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  oo        
____        
\   `       
 \       /x\
  \   sin|-|
   )     \n/
  /   ------
 /      n   
/___,       
n = 1

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

Sum(sin(x/n)/n, (n, 1, oo))

The radius of convergence of the power series

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

True

False

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/n)*sin(x/n)

The solution

Examples of finding the sum of a series