Mister Exam

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Sum of series:
(-1/2)^n
2(1/(n^2+5n+6))
(x+3)^n/2^n
((n+1)^n*3^(2n-1))/(2n+1)!
Identical expressions
n*i/ two ^(i+ one)
n multiply by i divide by 2 to the power of (i plus 1)
n multiply by i divide by two to the power of (i plus one)
n*i/2(i+1)
n*i/2i+1
ni/2^(i+1)
ni/2(i+1)
ni/2i+1
ni/2^i+1
n*i divide by 2^(i+1)
Similar expressions
n*i/2^(i-1)

Sum of series n*i/2^(i+1)

The solution

You have entered [src]

  oo        
____        
\   `       
 \     n*I  
  \   ------
  /    I + 1
 /    2     
/___,       
n = 0

$$\sum_{n=0}^{\infty} \frac{i n}{2^{1 + i}}$$

Sum((n*i)/2^(i + 1), (n, 0, oo))

The radius of convergence of the power series

Given number:
$$\frac{i n}{2^{1 + i}}$$
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
$$a_{n} = 2^{-1 - i} i n$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n}{n + 1}\right)$$
Let's take the limit
we find

True

False

The answer [src]

      -I
oo*I*2

$$\infty 2^{- i} i$$

oo*i*2^(-i)

Numerical answer

The series diverges

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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Identical expressions

Similar expressions

Sum of series n*i/2^(i+1)

The solution

Examples of finding the sum of a series