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Sum of series/ e^ipi/n/n

Sum of series e^ipi/n/n


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The solution

You have entered [src] 
  oo          
_____         
\    `        
 \     / I   \
  \    |E *pi|
   \   |-----|
   /   \  n  /
  /    -------
 /        n   
/____,        
n = 1
$$\sum_{n=1}^{\infty} \frac{\frac{1}{n} e^{i} \pi}{n}$$ 
Sum(((E^i*pi)/n)/n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\frac{1}{n} e^{i} \pi}{n}$$
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
$$a_{n} = \frac{\pi e^{i}}{n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n^{2}}\right)$$
Let's take the limit
we find
True

False
The answer [src] 
  3  I
pi *e 
------
  6
$$\frac{\pi^{3} e^{i}}{6}$$ 
pi^3*exp(i)/6
Numerical answer [src] 
2.79212713112525353446434904126 + 4.34848036223300095206604646324*i
2.79212713112525353446434904126 + 4.34848036223300095206604646324*i

    Examples of finding the sum of a series

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