Mister Exam

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Sum of series:
sin(sqrt(n^3-1)/(n^2+1))^2
((n^2)/(9n^2-1))^2
arctg(1/x^(1/3))
ln(n+1)/(n^(5/4))
Identical expressions
arctg(one /x^(one / three))
arctg(1 divide by x to the power of (1 divide by 3))
arctg(one divide by x to the power of (one divide by three))
arctg(1/x(1/3))
arctg1/x1/3
arctg1/x^1/3
arctg(1 divide by x^(1 divide by 3))

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

The solution

You have entered [src]

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

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

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

The radius of convergence of the power series

Given number:
$$\operatorname{atan}{\left(\frac{1}{\sqrt[3]{x}} \right)}$$
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
$$a_{n} = \operatorname{atan}{\left(\frac{1}{\sqrt[3]{x}} \right)}$$
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]

       /  1  \
oo*atan|-----|
       |3 ___|
       \\/ x /

$$\infty \operatorname{atan}{\left(\frac{1}{\sqrt[3]{x}} \right)}$$

oo*atan(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
```

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How to use it?

Identical expressions

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

The solution

Examples of finding the sum of a series