Mister Exam

Lang:

Other calculators

How to use it?
Sum of series:
5/pi^n
1/(4n^(2)-1)
factorial(n)/n^n
8^(n-1)/factorial(n-1)
Identical expressions
five /pi^n
5 divide by Pi to the power of n
five divide by Pi to the power of n
5/pin
5 divide by pi^n

Sum of series/ 5/pi^n

Sum of series 5/pi^n

The solution

You have entered [src]

  oo     
____     
\   `    
 \     5 
  \   ---
  /     n
 /    pi 
/___,    
n = 1

$$\sum_{n=1}^{\infty} \frac{5}{\pi^{n}}$$

Sum(5/pi^n, (n, 1, oo))

The radius of convergence of the power series

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

False

$$R = 0$$

The rate of convergence of the power series

The answer [src]

     5     
-----------
   /    1 \
pi*|1 - --|
   \    pi/

$$\frac{5}{\pi \left(1 - \frac{1}{\pi}\right)}$$

5/(pi*(1 - 1/pi))

Numerical answer [src]

2.33471103462129929991697406617

2.33471103462129929991697406617

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 5/pi^n

The solution

Examples of finding the sum of a series