Mister Exam

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Sum of series:
1/(n(n+1)(n+2))
1/3^n
1/(3n-2)(3n+1)
(x-1)^n/5^n
Identical expressions
cos(pi*n/(two *n+ five))
co sinus of e of ( Pi multiply by n divide by (2 multiply by n plus 5))
co sinus of e of ( Pi multiply by n divide by (two multiply by n plus five))
cos(pin/(2n+5))
cospin/2n+5
cos(pi*n divide by (2*n+5))
Similar expressions
cos(pi*n/(2n+5))
cos(pi*n/(2*n-5))

Sum of series/ cos(pi*n/(2*n+5))

Sum of series cos(pin/(2n+5))

The solution

You have entered [src]

  oo              
 ___              
 \  `             
  \      /  pi*n \
   )  cos|-------|
  /      \2*n + 5/
 /__,             
n = 1

$$\sum_{n=1}^{\infty} \cos{\left(\frac{\pi n}{2 n + 5} \right)}$$

Sum(cos((pi*n)/(2*n + 5)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\cos{\left(\frac{\pi n}{2 n + 5} \right)}$$
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
$$a_{n} = \cos{\left(\frac{\pi n}{2 n + 5} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\cos{\left(\frac{\pi n}{2 n + 5} \right)}}{\cos{\left(\frac{\pi \left(n + 1\right)}{2 n + 7} \right)}}}\right|$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

  oo              
 ___              
 \  `             
  \      /  pi*n \
   )  cos|-------|
  /      \5 + 2*n/
 /__,             
n = 1

$$\sum_{n=1}^{\infty} \cos{\left(\frac{\pi n}{2 n + 5} \right)}$$

Sum(cos(pi*n/(5 + 2*n)), (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
```

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How to use it?

Identical expressions

Similar expressions

Sum of series cos(pi*n/(2*n+5))

The solution

Examples of finding the sum of a series

Sum of series cos(pin/(2n+5))