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sin²n/√n³+5
Sum of series/ sin²n/√n³+5

Sum of series sin²n/√n³+5


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The solution

You have entered [src] 
  oo                
_____               
\    `              
 \     /   2       \
  \    |sin (n)    |
   \   |------- + 5|
   /   |      3    |
  /    |   ___     |
 /     \ \/ n      /
/____,              
n = 1
$$\sum_{n=1}^{\infty} \left(5 + \frac{\sin^{2}{\left(n \right)}}{\left(\sqrt{n}\right)^{3}}\right)$$ 
Sum(sin(n)^2/(sqrt(n))^3 + 5, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$5 + \frac{\sin^{2}{\left(n \right)}}{\left(\sqrt{n}\right)^{3}}$$
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
$$a_{n} = 5 + \frac{\sin^{2}{\left(n \right)}}{n^{\frac{3}{2}}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{5 + \frac{\sin^{2}{\left(n \right)}}{n^{\frac{3}{2}}}}{5 + \frac{\sin^{2}{\left(n + 1 \right)}}{\left(n + 1\right)^{\frac{3}{2}}}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo               
____               
\   `              
 \    /       2   \
  \   |    sin (n)|
   )  |5 + -------|
  /   |       3/2 |
 /    \      n    /
/___,              
n = 1
$$\sum_{n=1}^{\infty} \left(5 + \frac{\sin^{2}{\left(n \right)}}{n^{\frac{3}{2}}}\right)$$ 
Sum(5 + sin(n)^2/n^(3/2), (n, 1, oo))
The graph
Sum of series sin²n/√n³+5

    Examples of finding the sum of a series

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