Mister Exam

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Sum of series:
x^n/sqrt(n+1)
x^2j
3/n
1-cos(pi/n)
Identical expressions
x^n/sqrt(n+ one)
x to the power of n divide by square root of (n plus 1)
x to the power of n divide by square root of (n plus one)
x^n/√(n+1)
xn/sqrt(n+1)
xn/sqrtn+1
x^n/sqrtn+1
x^n divide by sqrt(n+1)
Similar expressions
x^n/sqrt(n-1)

Sum of series/ x^n/sqrt(n+1)

Sum of series x^n/sqrt(n+1)

The solution

You have entered [src]

  oo           
____           
\   `          
 \         n   
  \       x    
   )  ---------
  /     _______
 /    \/ n + 1 
/___,          
n = 0

\sum_{n=0}^{\infty} \frac{x^{n}}{\sqrt{n + 1}}

Sum(x^n/sqrt(n + 1), (n, 0, oo))

The radius of convergence of the power series

Given number:

\frac{x^{n}}{\sqrt{n + 1}}

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} = \frac{1}{\sqrt{n + 1}}

and

x_{0} = 0

,

d = 1

,

c = 1

then

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

Let's take the limit
we find

R = 1

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
```