Mister Exam

Lang:

Sum of series (2i+1)/n^2

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  oo         
____         
\   `        
 \    2*I + 1
  \   -------
  /       2  
 /       n   
/___,        
n = 1

\sum_{n=1}^{\infty} \frac{1 + 2 i}{n^{2}}

Sum((2*i + 1)/n^2, (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{1 + 2 i}{n^{2}}

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{1 + 2 i}{n^{2}}

and

x_{0} = 0

d = 0

c = 1

then

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

True

False

The answer [src]

  2       2
pi    I*pi 
--- + -----
 6      3

\frac{\pi^{2}}{6} + \frac{i \pi^{2}}{3}

pi^2/6 + i*pi^2/3

Numerical answer [src]

1.64493406684822643647241516665 + 3.28986813369645287294483033329*i

1.64493406684822643647241516665 + 3.28986813369645287294483033329*i