Mister Exam

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Sum of series:
1/n^n
1/((3*n+1)*(3*n+4))
(2n+1)/(n(n+2)(n+4))
1/(2*n-1)*(2*n+1)
Identical expressions
one /((x(x+ one)))^(one / two)
1 divide by ((x(x plus 1))) to the power of (1 divide by 2)
one divide by ((x(x plus one))) to the power of (one divide by two)
1/((x(x+1)))(1/2)
1/xx+11/2
1/xx+1^1/2
1 divide by ((x(x+1)))^(1 divide by 2)
Similar expressions
1/((x(x-1)))^(1/2)

Sum of series 1/((x(x+1)))^(1/2)

The solution

You have entered [src]

  oo               
____               
\   `              
 \          1      
  \   -------------
  /     ___________
 /    \/ x*(x + 1) 
/___,              
n = 1

\sum_{n=1}^{\infty} \frac{1}{\sqrt{x \left(x + 1\right)}}

Sum(1/(sqrt(x*(x + 1))), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{1}{\sqrt{x \left(x + 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} = \frac{1}{\sqrt{x \left(x + 1\right)}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty} 1

Let's take the limit
we find

True

False

The answer [src]

      oo     
-------------
  ___________
\/ x*(1 + x)

\frac{\infty}{\sqrt{x \left(x + 1\right)}}

oo/sqrt(x*(1 + 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
```

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How to use it?

Identical expressions

Similar expressions

Sum of series 1/((x(x+1)))^(1/2)

The solution

Examples of finding the sum of a series