Mister Exam

Other calculators

Sum of series/ x^n/(√(n+1)*4n)

Sum of series x^n/(√(n+1)*4n)


∑
=

The solution

You have entered [src] 
  oo               
____               
\   `              
 \           n     
  \         x      
   )  -------------
  /     _______    
 /    \/ n + 1 *4*n
/___,              
n = 0
$$\sum_{n=0}^{\infty} \frac{x^{n}}{n 4 \sqrt{n + 1}}$$ 
Sum(x^n/(((sqrt(n + 1)*4)*n)), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{x^{n}}{n 4 \sqrt{n + 1}}$$
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
$$a_{n} = \frac{1}{4 n \sqrt{n + 1}}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{\sqrt{n + 1} \sqrt{n + 2}}{n}\right)$$
Let's take the limit
we find
$$R = 1$$
The answer [src] 
  oo               
____               
\   `              
 \           n     
  \         x      
   )  -------------
  /         _______
 /    4*n*\/ 1 + n 
/___,              
n = 0
$$\sum_{n=0}^{\infty} \frac{x^{n}}{4 n \sqrt{n + 1}}$$ 
Sum(x^n/(4*n*sqrt(1 + n)), (n, 0, oo))

    Examples of finding the sum of a series

    © Mister Exam – Calculator