Mister Exam

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Sum of series:
sqrt(n)-2*sqrt(n+1)+sqrt(n+2)
sqrt(1+1)/1
x^n*4^n/((5^n*(n+9)^(1/9)))
(157/50)^2*259/(4*100)
Identical expressions
arcsin(n+ one /n^ two)
arc sinus of (n plus 1 divide by n squared )
arc sinus of (n plus one divide by n to the power of two)
arcsin(n+1/n2)
arcsinn+1/n2
arcsin(n+1/n²)
arcsin(n+1/n to the power of 2)
arcsinn+1/n^2
arcsin(n+1 divide by n^2)
Similar expressions
arcsin(n-1/n^2)

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

The solution

You have entered [src]

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

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

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

The radius of convergence of the power series

Given number:

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

and

x_{0} = 0

d = 0

c = 1

then

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

Let's take the limit
we find

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

False

The answer [src]

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

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

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

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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Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series