Mister Exam

# Sum of series sin(pi*n/4)/ln(n)

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### The solution

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  oo
____
\
\       /pi*n\
\   sin|----|
)     \ 4  /
/   ---------
/      log(n)
/___,
n = 1          
$$\sum_{n=1}^{\infty} \frac{\sin{\left(\frac{\pi n}{4} \right)}}{\log{\left(n \right)}}$$
Sum(sin((pi*n)/4)/log(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(\frac{\pi n}{4} \right)}}{\log{\left(n \right)}}$$
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
$$a_{n} = \frac{\sin{\left(\frac{\pi n}{4} \right)}}{\log{\left(n \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\log{\left(n + 1 \right)} \left|{\frac{\sin{\left(\frac{\pi n}{4} \right)}}{\log{\left(n \right)} \sin{\left(\pi \left(\frac{n}{4} + \frac{1}{4}\right) \right)}}}\right|\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\log{\left(n + 1 \right)} \left|{\frac{\sin{\left(\frac{\pi n}{4} \right)}}{\log{\left(n \right)} \sin{\left(\pi \left(\frac{n}{4} + \frac{1}{4}\right) \right)}}}\right|\right)$$
False
The rate of convergence of the power series
  oo
____
\   
\       /pi*n\
\   sin|----|
)     \ 4  /
/   ---------
/      log(n)
/___,
n = 1          
$$\sum_{n=1}^{\infty} \frac{\sin{\left(\frac{\pi n}{4} \right)}}{\log{\left(n \right)}}$$
Sum(sin(pi*n/4)/log(n), (n, 1, oo))
The graph
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