Mister Exam

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How to use it?
Sum of series:
t^k*(-1)^k*a^(2*k)*cos(a*x)*e^(a*z)/factorial(k)
(-1)^n*(2*2)^(2*n)/factorial(2*n)
sqrt(n)-2*sqrt(n+1)+sqrt(n+2)
arcctg((n+1)/(n^2-3))
Graphing y =:
1/(x(x+1))
Integral of d{x}:
1/(x(x+1))
Identical expressions
one /(x(x+ one))
1 divide by (x(x plus 1))
one divide by (x(x plus one))
1/xx+1
1 divide by (x(x+1))
Similar expressions
1/(x(x-1))
1/(x*(x+1)*(x+2))

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

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

The solution

You have entered [src]

  oo           
 ___           
 \  `          
  \       1    
   )  ---------
  /   x*(x + 1)
 /__,          
k = 1

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

Sum(1/(x*(x + 1)), (k, 1, oo))

The radius of convergence of the power series

Given number:

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

It is a series of species

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

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

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

In this case

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

and

x_{0} = 0

,

d = 0

,

c = 1

then

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

Let's take the limit
we find

True

False

The answer [src]

    oo   
---------
x*(1 + x)

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

oo/(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
```