Mister Exam

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How to use it?
Sum of series:
1/(2n+1)
(n-1)/factorial(n)
5/((5n-2)*(5n+3))
k/2^k
Identical expressions
k/ two ^k
k divide by 2 to the power of k
k divide by two to the power of k
k/2k
k divide by 2^k

Sum of series/ k/2^k

Sum of series k/2^k

The solution

You have entered [src]

  oo    
____    
\   `   
 \    k 
  \   --
  /    k
 /    2 
/___,   
n = 1

\sum_{n=1}^{\infty} \frac{k}{2^{k}}

Sum(k/2^k, (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{k}{2^{k}}

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} = 2^{- k} k

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]

      -k
oo*k*2

\infty 2^{- k} k

oo*k*2^(-k)

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
```