Sum of series nln(n)/e^n

The solution

You have entered [src]

  oo          
____          
\   `         
 \    n*log(n)
  \   --------
  /       n   
 /       E    
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{n \log{\left(n \right)}}{e^{n}}$$

Sum((n*log(n))/E^n, (n, 1, oo))

The radius of convergence of the power series

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

False

$$R = 0$$

The rate of convergence of the power series

The answer [src]

  oo              
 ___              
 \  `             
  \      -n       
  /   n*e  *log(n)
 /__,             
n = 1

$$\sum_{n=1}^{\infty} n e^{- n} \log{\left(n \right)}$$

Sum(n*exp(-n)*log(n), (n, 1, oo))

Numerical answer [src]

0.556375615722430028502747844401

0.556375615722430028502747844401

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
```

Other calculators

How to use it?

Identical expressions

Sum of series nln(n)/e^n

The solution

Examples of finding the sum of a series