Mister Exam

Lang:

Other calculators

How to use it?
Sum of series:
5/n
(n-2^(3*n))/(n+1)^(1/3)
((3*x)^n)/n!
1/((5n-4)(5n+1))
Identical expressions
arcsin(n/(n^ two + one))
arc sinus of (n divide by (n squared plus 1))
arc sinus of (n divide by (n to the power of two plus one))
arcsin(n/(n2+1))
arcsinn/n2+1
arcsin(n/(n²+1))
arcsin(n/(n to the power of 2+1))
arcsinn/n^2+1
arcsin(n divide by (n^2+1))
Similar expressions
arcsin(n/(n^2-1))

Sum of series/ arcsin(n/(n^2+1))

Sum of series arcsin(n/(n^2+1))

The solution

You have entered [src]

  oo              
____              
\   `             
 \        /  n   \
  \   asin|------|
  /       | 2    |
 /        \n  + 1/
/___,             
n = 1

\sum_{n=1}^{\infty} \operatorname{asin}{\left(\frac{n}{n^{2} + 1} \right)}

Sum(asin(n/(n^2 + 1)), (n, 1, oo))

The radius of convergence of the power series

Given number:

\operatorname{asin}{\left(\frac{n}{n^{2} + 1} \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{asin}{\left(\frac{n}{n^{2} + 1} \right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty} \left|{\frac{\operatorname{asin}{\left(\frac{n}{n^{2} + 1} \right)}}{\operatorname{asin}{\left(\frac{n + 1}{\left(n + 1\right)^{2} + 1} \right)}}}\right|

Let's take the limit
we find

True

False

The rate of convergence of the power series

The graph

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

Similar expressions

Sum of series arcsin(n/(n^2+1))

The solution

Examples of finding the sum of a series