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  • Sum of series:
  • n^2/3^n n^2/3^n
  • nx^n/(8n-9)*3^n
  • cos(i*n)/2^n cos(i*n)/2^n
  • sin(3*n)/(n+2) sin(3*n)/(n+2)

  • Identical expressions

  • arcsin(pi/(n^ two + two))
  • arc sinus of ( Pi divide by (n squared plus 2))
  • arc sinus of ( Pi divide by (n to the power of two plus two))
  • arcsin(pi/(n2+2))
  • arcsinpi/n2+2
  • arcsin(pi/(n²+2))
  • arcsin(pi/(n to the power of 2+2))
  • arcsinpi/n^2+2
  • arcsin(pi divide by (n^2+2))

  • Similar expressions

  • arcsin(pi/(n^2-2))
Sum of series/ arcsin(pi/(n^2+2))

Sum of series arcsin(pi/(n^2+2))


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The solution

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