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

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

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

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

The radius of convergence of the power series

Given number:

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

and

x_{0} = 0

d = 0

c = 1

then

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

True

False

The rate of convergence of the power series

The graph