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  • Sum of series:
  • (-1)^n/2n-1 (-1)^n/2n-1
  • 2^n 2^n
  • (x-1)^n/3^n
  • ln(n+1)/(n+1) ln(n+1)/(n+1)

  • Identical expressions

  • sin(x/n^ two)
  • sinus of (x divide by n squared )
  • sinus of (x divide by n to the power of two)
  • sin(x/n2)
  • sinx/n2
  • sin(x/n²)
  • sin(x/n to the power of 2)
  • sinx/n^2
  • sin(x divide by n^2)
Sum of series/ sin(x/n^2)

Sum of series sin(x/n^2)


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The solution

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