Mister Exam

Other calculators


1/ln^n(n+1)

Sum of series 1/ln^n(n+1)



=

The solution

You have entered [src]
  oo             
____             
\   `            
 \         1     
  \   -----------
  /      n       
 /    log (n + 1)
/___,            
n = 1            
$$\sum_{n=1}^{\infty} \frac{1}{\log{\left(n + 1 \right)}^{n}}$$
Sum(1/(log(n + 1)^n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\log{\left(n + 1 \right)}^{n}}$$
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
$$a_{n} = \log{\left(n + 1 \right)}^{- n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\log{\left(n + 1 \right)}^{- n} \log{\left(n + 2 \right)}^{n + 1}\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src]
  oo              
 ___              
 \  `             
  \      -n       
  /   log  (1 + n)
 /__,             
n = 1             
$$\sum_{n=1}^{\infty} \log{\left(n + 1 \right)}^{- n}$$
Sum(log(1 + n)^(-n), (n, 1, oo))
Numerical answer [src]
2.87674663155412234729191448477
2.87674663155412234729191448477
The graph
Sum of series 1/ln^n(n+1)

    Examples of finding the sum of a series