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sin*(2n+1/(n^3))

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  • Identical expressions

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  • Similar expressions

  • sin*(2n-1/(n^3))
Sum of series/ sin*(2n+1/(n^3))

Sum of series sin*(2n+1/(n^3))


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The solution

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

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