Mister Exam

Lang:

Sum of series/ (-1)ln(n+2/n)

Sum of series (-1)ln(n+2/n)

You have entered [src]

  oo             
 ___             
 \  `            
  \       /    2\
   )  -log|n + -|
  /       \    n/
 /__,            
n = 1

\sum_{n=1}^{\infty} - \log{\left(n + \frac{2}{n} \right)}

Sum(-log(n + 2/n), (n, 1, oo))

The radius of convergence of the power series

Given number:

- \log{\left(n + \frac{2}{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} = - \log{\left(n + \frac{2}{n} \right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left|{\log{\left(n + \frac{2}{n} \right)}}\right|}{\log{\left(n + 1 + \frac{2}{n + 1} \right)}}\right)

1 = \lim_{n \to \infty}\left(\frac{\left|{\log{\left(n + \frac{2}{n} \right)}}\right|}{\log{\left(n + 1 + \frac{2}{n + 1} \right)}}\right)

False

The rate of convergence of the power series

The graph