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Sum of series:
x^n/n
1/n^6
1/n^n
(n-1)/n!
Identical expressions
one /(one * three)+ one /((2n- one)*(2n+ one))
1 divide by (1 multiply by 3) plus 1 divide by ((2n minus 1) multiply by (2n plus 1))
one divide by (one multiply by three) plus one divide by ((2n minus one) multiply by (2n plus one))
1/(13)+1/((2n-1)(2n+1))
1/13+1/2n-12n+1
1 divide by (1*3)+1 divide by ((2n-1)*(2n+1))
Similar expressions
1/(1*3)-1/((2n-1)*(2n+1))
1/(1*3)+1/((2n-1)*(2n-1))
1/(1*3)+1/((2n+1)*(2n+1))

Sum of series/ 1/(1*3)+1/((2n-1)*(2n+1))

Sum of series 1/(13)+1/((2n-1)(2n+1))

The solution

You have entered [src]

  oo                                           
 ___                                           
 \  `                                          
  \   /                             1         \
   )  |0.333333333333333 + -------------------|
  /   \                    (2*n - 1)*(2*n + 1)/
 /__,                                          
n = 1

$$\sum_{n=1}^{\infty} \left(0.333333333333333 + \frac{1}{\left(2 n - 1\right) \left(2 n + 1\right)}\right)$$

Sum(0.333333333333333 + 1/((2*n - 1)*(2*n + 1)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$0.333333333333333 + \frac{1}{\left(2 n - 1\right) \left(2 n + 1\right)}$$
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
$$a_{n} = 0.333333333333333 + \frac{1}{\left(2 n - 1\right) \left(2 n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{0.333333333333333 + \frac{1}{\left(2 n - 1\right) \left(2 n + 1\right)}}\right|}{0.333333333333333 + \frac{1}{\left(2 n + 1\right) \left(2 n + 3\right)}}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

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$$\infty$$

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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

Similar expressions

Sum of series 1/(1*3)+1/((2n-1)*(2n+1))

The solution

Examples of finding the sum of a series

Sum of series 1/(13)+1/((2n-1)(2n+1))