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Sum of series/ sin(1/n^2)

Sum of series sin(1/n^2)

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  oo         
____         
\   `        
 \       /1 \
  \   sin|--|
  /      | 2|
 /       \n /
/___,        
n = 1

\sum_{n=1}^{\infty} \sin{\left(\frac{1}{n^{2}} \right)}

Sum(sin(1/(n^2)), (n, 1, oo))

The radius of convergence of the power series

Given number:

\sin{\left(\frac{1}{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{1}{n^{2}} \right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{1}{n^{2}} \right)}}{\sin{\left(\frac{1}{\left(n + 1\right)^{2}} \right)}}}\right|

True

False

The rate of convergence of the power series

The answer [src]

  oo         
____         
\   `        
 \       /1 \
  \   sin|--|
  /      | 2|
 /       \n /
/___,        
n = 1

\sum_{n=1}^{\infty} \sin{\left(\frac{1}{n^{2}} \right)}

Sum(sin(n^(-2)), (n, 1, oo))

Numerical answer [src]

1.48352281730955228641929918636

1.48352281730955228641929918636

The graph