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Sum of series:
5^n
24/((5-3n)(2-3n))
0.5^n
(-1)/(2n-1)!
Identical expressions
(arcsin(x)/x)^ctg(x)
(arc sinus of (x) divide by x) to the power of ctg(x)
(arcsin(x)/x)ctg(x)
arcsinx/xctgx
arcsinx/x^ctgx
(arcsin(x) divide by x)^ctg(x)
Similar expressions
(arcsinx/x)^ctg(x)

Sum of series (arcsin(x)/x)^ctg(x)

The solution

You have entered [src]

  oo                 
____                 
\   `                
 \             cot(x)
  \   /asin(x)\      
  /   |-------|      
 /    \   x   /      
/___,                
n = 1

\sum_{n=1}^{\infty} \left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{\cot{\left(x \right)}}

Sum((asin(x)/x)^cot(x), (n, 1, oo))

The radius of convergence of the power series

Given number:

\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{\cot{\left(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} = \left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{\cot{\left(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]

            cot(x)
   /asin(x)\      
oo*|-------|      
   \   x   /

\infty \left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{\cot{\left(x \right)}}

oo*(asin(x)/x)^cot(x)

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 (arcsin(x)/x)^ctg(x)

The solution

Examples of finding the sum of a series