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sin(1/n^2)

  • How to use it?

  • Sum of series:
  • (5^n-2^n)/10^n (5^n-2^n)/10^n
  • sin(1/n^2) sin(1/n^2)
  • factorial(x-n)
  • (n-1)*5 (n-1)*5

  • Identical expressions

  • sin(one /n^ two)
  • sinus of (1 divide by n squared )
  • sinus of (one divide by n to the power of two)
  • sin(1/n2)
  • sin1/n2
  • sin(1/n²)
  • sin(1/n to the power of 2)
  • sin1/n^2
  • sin(1 divide by n^2)
Sum of series/ sin(1/n^2)

Sum of series sin(1/n^2)


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The solution

You have entered [src] 
  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
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{1}{n^{2}} \right)}}{\sin{\left(\frac{1}{\left(n + 1\right)^{2}} \right)}}}\right|$$
Let's take the limit
we find
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
Sum of series sin(1/n^2)

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