Mister Exam

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Sum of series:
24/(9n^2-12n-5)
(3^(n+2)-2*6^n)/18^n
(-1)^nx^n/n!
((2*n+4)/(5*n+7))^n*x^n
Identical expressions
sqrt(n/(two n+2))
square root of (n divide by (2n plus 2))
square root of (n divide by (two n plus 2))
√(n/(2n+2))
sqrtn/2n+2
sqrt(n divide by (2n+2))
Similar expressions
sqrt(n/(2n-2))

Sum of series/ sqrt(n/(2n+2))

Sum of series sqrt(n/(2n+2))

The solution

You have entered [src]

  oo               
____               
\   `              
 \        _________
  \      /    n    
  /     /  ------- 
 /    \/   2*n + 2 
/___,              
n = 1

\sum_{n=1}^{\infty} \sqrt{\frac{n}{2 n + 2}}

Sum(sqrt(n/(2*n + 2)), (n, 1, oo))

The radius of convergence of the power series

Given number:

\sqrt{\frac{n}{2 n + 2}}

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} = \sqrt{\frac{n}{2 n + 2}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\sqrt{n} \sqrt{2 n + 4}}{\sqrt{n + 1} \sqrt{2 n + 2}}\right)

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

  oo             
____             
\   `            
 \         ___   
  \      \/ n    
   )  -----------
  /     _________
 /    \/ 2 + 2*n 
/___,            
n = 1

\sum_{n=1}^{\infty} \frac{\sqrt{n}}{\sqrt{2 n + 2}}

Sum(sqrt(n)/sqrt(2 + 2*n), (n, 1, oo))

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
```

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How to use it?

Identical expressions

Similar expressions

Sum of series sqrt(n/(2n+2))

The solution

Examples of finding the sum of a series