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Sum of series:
1/(n*(n+1))
1/n^p
nx^(n!)
1/((2*n+5)*(2*n+7))
Derivative of:
1/xsin(x)
Integral of d{x}:
1/xsin(x)
Identical expressions
one /xsin(x)
1 divide by x sinus of (x)
one divide by x sinus of (x)
1/xsinx
1 divide by xsin(x)
Similar expressions
1/xsinx

Sum of series/ 1/xsin(x)

Sum of series 1/xsin(x)

The solution

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  oo        
 ___        
 \  `       
  \   sin(x)
   )  ------
  /     x   
 /__,       
x = 1

$$\sum_{x=1}^{\infty} \frac{\sin{\left(x \right)}}{x}$$

Sum(sin(x)/x, (x, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\sin{\left(x \right)}}{x}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \frac{\sin{\left(x \right)}}{x}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{\left(x + 1\right) \left|{\frac{\sin{\left(x \right)}}{\sin{\left(x + 1 \right)}}}\right|}{x}\right)$$
Let's take the limit
we find

True

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
```

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How to use it?

Identical expressions

Similar expressions

Sum of series 1/xsin(x)

The solution

Examples of finding the sum of a series