Mister Exam

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How to use it?
Sum of series:
x^n/n
3/(n(n+2))
n^2*sin(5/(3^n))
n*2^n
Identical expressions
cos(two *n)/(n*n^(one / two))
co sinus of e of (2 multiply by n) divide by (n multiply by n to the power of (1 divide by 2))
co sinus of e of (two multiply by n) divide by (n multiply by n to the power of (one divide by two))
cos(2*n)/(n*n(1/2))
cos2*n/n*n1/2
cos(2n)/(nn^(1/2))
cos(2n)/(nn(1/2))
cos2n/nn1/2
cos2n/nn^1/2
cos(2*n) divide by (n*n^(1 divide by 2))

Sum of series/ cos(2*n)/(n*n^(1/2))

Sum of series cos(2n)/(nn^(1/2))

The solution

You have entered [src]

  oo          
____          
\   `         
 \    cos(2*n)
  \   --------
  /       ___ 
 /    n*\/ n  
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{\cos{\left(2 n \right)}}{\sqrt{n} n}$$

Sum(cos(2*n)/((n*sqrt(n))), (n, 1, oo))

The radius of convergence of the power series

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

False

The rate of convergence of the power series

The answer [src]

  oo          
____          
\   `         
 \    cos(2*n)
  \   --------
  /      3/2  
 /      n     
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{\cos{\left(2 n \right)}}{n^{\frac{3}{2}}}$$

Sum(cos(2*n)/n^(3/2), (n, 1, oo))

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
```

Other calculators

How to use it?

Identical expressions

Sum of series cos(2*n)/(n*n^(1/2))

The solution

Examples of finding the sum of a series

Sum of series cos(2n)/(nn^(1/2))