Mister Exam

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Sum of series:
1/n^2
cos(n)/n
ln(k+1/k+2)
e^n/(1+e^(2*n))
Graphing y =:
cos(n)/n
Limit of the function:
cos(n)/n
Identical expressions
cos(n)/n
co sinus of e of (n) divide by n
cosn/n
cos(n) divide by n
Similar expressions
cos(n)/(n^(2/3)+1)
(n+cosn)/((n^2)(sqrt(n)))
(1-cosn)/(n^2+n-1)

Sum of series/ cos(n)/n

Sum of series cos(n)/n

The solution

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  oo        
 ___        
 \  `       
  \   cos(n)
   )  ------
  /     n   
 /__,       
n = 1

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

Sum(cos(n)/n, (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\cos{\left(n \right)}}{n}$$
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
$$a_{n} = \frac{\cos{\left(n \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\cos{\left(n \right)}}{\cos{\left(n + 1 \right)}}}\right|}{n}\right)$$
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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How to use it?

Identical expressions

Similar expressions

Sum of series cos(n)/n

The solution

Examples of finding the sum of a series