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Sum of series/ log(1-1/n^2)

Sum of series log(1-1/n^2)

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  oo             
____             
\   `            
 \       /    1 \
  \   log|1 - --|
  /      |     2|
 /       \    n /
/___,            
n = 2

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

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

The radius of convergence of the power series

Given number:

\log{\left(1 - \frac{1}{n^{2}} \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(1 - \frac{1}{n^{2}} \right)}

and

x_{0} = 0

d = 0

c = 1

then

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

True

False

The rate of convergence of the power series

Numerical answer [src]

-0.693147180559945309417232121458

-0.693147180559945309417232121458

The graph