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tan(pi/(4*n))
Sum of series/ tan(pi/(4*n))

Sum of series tan(pi/(4*n))


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The solution

You have entered [src] 
  oo          
 ___          
 \  `         
  \      / pi\
   )  tan|---|
  /      \4*n/
 /__,         
n = 1
$$\sum_{n=1}^{\infty} \tan{\left(\frac{\pi}{4 n} \right)}$$ 
Sum(tan(pi/((4*n))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\tan{\left(\frac{\pi}{4 n} \right)}$$
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
$$a_{n} = \tan{\left(\frac{\pi}{4 n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\tan{\left(\frac{\pi}{4 n} \right)}}{\tan{\left(\frac{\pi}{4 \left(n + 1\right)} \right)}}}\right|$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo          
 ___          
 \  `         
  \      / pi\
   )  tan|---|
  /      \4*n/
 /__,         
n = 1
$$\sum_{n=1}^{\infty} \tan{\left(\frac{\pi}{4 n} \right)}$$ 
Sum(tan(pi/(4*n)), (n, 1, oo))
The graph
Sum of series tan(pi/(4*n))

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