Mister Exam

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Sum of series:
1/(2^n)
(1+n)/(1+n^2)
(-1)^n/sqrt(n)
1/4n^2+1
Derivative of:
x^(-1/3)
Limit of the function:
x^(-1/3)
Integral of d{x}:
x^(-1/3)
Identical expressions
x^(- one / three)
x to the power of ( minus 1 divide by 3)
x to the power of ( minus one divide by three)
x(-1/3)
x-1/3
x^-1/3
x^(-1 divide by 3)
Similar expressions
x^(1/3)

Sum of series/ x^(-1/3)

Sum of series x^(-1/3)

The solution

You have entered [src]

  oo       
____       
\   `      
 \      1  
  \   -----
  /   3 ___
 /    \/ x 
/___,      
n = 1

\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{x}}

Sum(x^(-1/3), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{1}{\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{1}{\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]

  oo 
-----
3 ___
\/ x

\frac{\infty}{\sqrt[3]{x}}

oo/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
```