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Sum of series:
24/(9n^2-12n-5)
t^k*(-1)^k*a^(2*k)*cos(a*x)*e^(a*z)/factorial(k)
factorial(n)/(2^n+1)
x^n*4^n/((5^n*(n+9)^(1/9)))
Identical expressions
sin^ two (x)/(x^(one / three))
sinus of squared (x) divide by (x to the power of (1 divide by 3))
sinus of to the power of two (x) divide by (x to the power of (one divide by three))
sin2(x)/(x(1/3))
sin2x/x1/3
sin²(x)/(x^(1/3))
sin to the power of 2(x)/(x to the power of (1/3))
sin^2x/x^1/3
sin^2(x) divide by (x^(1 divide by 3))

Sum of series sin^2(x)/(x^(1/3))

The solution

You have entered [src]

  oo         
____         
\   `        
 \       2   
  \   sin (x)
   )  -------
  /    3 ___ 
 /     \/ x  
/___,        
n = 1

$$\sum_{n=1}^{\infty} \frac{\sin^{2}{\left(x \right)}}{\sqrt[3]{x}}$$

Sum(sin(x)^2/x^(1/3), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\sin^{2}{\left(x \right)}}{\sqrt[3]{x}}$$
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
$$a_{n} = \frac{\sin^{2}{\left(x \right)}}{\sqrt[3]{x}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find

True

False

The answer [src]

      2   
oo*sin (x)
----------
  3 ___   
  \/ x

$$\frac{\infty \sin^{2}{\left(x \right)}}{\sqrt[3]{x}}$$

oo*sin(x)^2/x^(1/3)

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(x)/(x^(1/3))

The solution

Examples of finding the sum of a series