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Sum of series:
sqrt(n)*(n/(4*n-3))^(2*n)
tan(n*x)
arctan(n)/(n!)^2
ln(n)/sqrt(n)
Identical expressions
sinx/(n(n)^(one / three))
sinus of x divide by (n(n) to the power of (1 divide by 3))
sinus of x divide by (n(n) to the power of (one divide by three))
sinx/(n(n)(1/3))
sinx/nn1/3
sinx/nn^1/3
sinx divide by (n(n)^(1 divide by 3))

Sum of series sinx/(n(n)^(1/3))

The solution

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  oo         
____         
\   `        
 \     sin(x)
  \   -------
  /     3 ___
 /    n*\/ n 
/___,        
n = 1

$$\sum_{n=1}^{\infty} \frac{\sin{\left(x \right)}}{\sqrt[3]{n} n}$$

Sum(sin(x)/((n*n^(1/3))), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\sin{\left(x \right)}}{\sqrt[3]{n} n}$$
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
$$a_{n} = \frac{\sin{\left(x \right)}}{n^{\frac{4}{3}}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{\frac{4}{3}}}{n^{\frac{4}{3}}}\right)$$
Let's take the limit
we find

True

False

The answer [src]

  oo        
____        
\   `       
 \    sin(x)
  \   ------
  /     4/3 
 /     n    
/___,       
n = 1

$$\sum_{n=1}^{\infty} \frac{\sin{\left(x \right)}}{n^{\frac{4}{3}}}$$

Sum(sin(x)/n^(4/3), (n, 1, oo))

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 sinx/(n(n)^(1/3))

The solution

Examples of finding the sum of a series