Mister Exam

Other calculators

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

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


∑
=

The solution

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

False
The rate of convergence of the power series
Numerical answer
The series diverges
The graph
Sum of series n*ln(1+1/n)

    Examples of finding the sum of a series

    © Mister Exam – Calculator