Sum of series sin^2(n)

The solution

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  oo         
 ___         
 \  `        
  \      2   
  /   sin (n)
 /__,        
n = 1

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

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

The radius of convergence of the power series

Given number:
$$\sin^{2}{\left(n \right)}$$
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
$$a_{n} = \sin^{2}{\left(n \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\sin^{2}{\left(n \right)} \left|{\frac{1}{\sin^{2}{\left(n + 1 \right)}}}\right|\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\sin^{2}{\left(n \right)} \left|{\frac{1}{\sin^{2}{\left(n + 1 \right)}}}\right|\right)$$

False

The rate of convergence of the power series

Numerical answer

The series diverges

The graph

Examples of finding the sum of a series

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

Other calculators

How to use it?

Identical expressions

Sum of series sin^2(n)

The solution

Examples of finding the sum of a series