Mister Exam

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Sum of series:
n^2*sin(5/(3^n))
n*2^n
n^3
(2n+1)/(n(n+2)(n+4))
Limit of the function:
n*2^n
Identical expressions
n* two ^n
n multiply by 2 to the power of n
n multiply by two to the power of n
n*2n
n2^n
n2n

Sum of series/ n*2^n

Sum of series n*2^n

The solution

You have entered [src]

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 \  `     
  \      n
  /   n*2 
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n = 1

$$\sum_{n=1}^{\infty} 2^{n} n$$

Sum(n*2^n, (n, 1, oo))

The radius of convergence of the power series

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

False

The rate of convergence of the power series

The answer [src]

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$$\infty$$

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Numerical answer

The series diverges

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

Sum of series n*2^n

The solution

Examples of finding the sum of a series