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Sum of series/ (xi)2

Sum of series (xi)2

The solution

You have entered [src]

  oo       
 __        
 \ `       
  )   x*i*2
 /_,       
i = 3

\sum_{i=3}^{\infty} 2 i x

Sum((x*i)*2, (i, 3, oo))

The radius of convergence of the power series

Given number:

2 i x

It is a series of species

a_{i} \left(c x - x_{0}\right)^{d i}

- power series.
The radius of convergence of a power series can be calculated by the formula:

R^{d} = \frac{x_{0} + \lim_{i \to \infty} \left|{\frac{a_{i}}{a_{i + 1}}}\right|}{c}

In this case

a_{i} = 2 i x

and

x_{0} = 0

,

d = 0

,

c = 1

then

1 = \lim_{i \to \infty}\left(\frac{i}{i + 1}\right)

Let's take the limit
we find

True

False

The answer [src]

oo*x

\infty x

oo*x

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
```