Mister Exam

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Sum of series:
x^n/(1-x^n)
1/((2*n-1)(2*n+1))
arcsin(1/sqrt(n))
sin*(2n+1/(n^3))
Identical expressions
one /((n(n+ one)(n+ two)(n+ three)))
1 divide by ((n(n plus 1)(n plus 2)(n plus 3)))
one divide by ((n(n plus one)(n plus two)(n plus three)))
1/nn+1n+2n+3
1 divide by ((n(n+1)(n+2)(n+3)))
Similar expressions
1/((n(n-1)(n+2)(n+3)))
1/((n(n+1)(n+2)(n-3)))
1/((n(n+1)(n-2)(n+3)))

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

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

The solution

You have entered [src]

  oo                           
 ___                           
 \  `                          
  \               1            
   )  -------------------------
  /   n*(n + 1)*(n + 2)*(n + 3)
 /__,                          
n = 1

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

Sum(1/(((n*(n + 1))*(n + 2))*(n + 3)), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\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} = \frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{n + 4}{n}\right)

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

1/18

\frac{1}{18}

1/18

Numerical answer [src]

0.0555555555555555555555555555556

0.0555555555555555555555555555556

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 1/((n(n+1)(n+2)(n+3)))

The solution

Examples of finding the sum of a series