Mister Exam

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Sum of series:
1/(2n+1)
(n-1)/factorial(n)
5/((5n-2)*(5n+3))
ln(k+1)/n
Identical expressions
five /((5n- two)*(5n+ three))
5 divide by ((5n minus 2) multiply by (5n plus 3))
five divide by ((5n minus two) multiply by (5n plus three))
5/((5n-2)(5n+3))
5/5n-25n+3
5 divide by ((5n-2)*(5n+3))
Similar expressions
5/((5n+2)*(5n+3))
5/((5n-2)(5n+3))
5/((5n-2)*(5n-3))
5/(5n-2)(5n+3)

Sum of series/ 5/((5n-2)*(5n+3))

Sum of series 5/((5n-2)*(5n+3))

The solution

You have entered [src]

  oo                     
 ___                     
 \  `                    
  \            5         
   )  -------------------
  /   (5*n - 2)*(5*n + 3)
 /__,                    
n = 1

\sum_{n=1}^{\infty} \frac{5}{\left(5 n - 2\right) \left(5 n + 3\right)}

Sum(5/(((5*n - 2)*(5*n + 3))), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{5}{\left(5 n - 2\right) \left(5 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{5}{\left(5 n - 2\right) \left(5 n + 3\right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\left(5 n + 8\right) \left|{\frac{1}{5 n - 2}}\right|\right)

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

5*Gamma(13/5)
-------------
24*Gamma(8/5)

\frac{5 \Gamma\left(\frac{13}{5}\right)}{24 \Gamma\left(\frac{8}{5}\right)}

5*gamma(13/5)/(24*gamma(8/5))

Numerical answer [src]

0.333333333333333333333333333333

0.333333333333333333333333333333

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 5/((5n-2)*(5n+3))

The solution

Examples of finding the sum of a series