Mister Exam

Other calculators

  • How to use it?

  • Sum of series:
  • atan(1+pi/2)^n atan(1+pi/2)^n
  • seno^2x(2x)
  • n/2^n*z^2n
  • log_{e}((n(n+2))/(n+1)^2) log_{e}((n(n+2))/(n+1)^2)

  • Identical expressions

  • n/ two ^n*z^2n
  • n divide by 2 to the power of n multiply by z squared n
  • n divide by two to the power of n multiply by z squared n
  • n/2n*z2n
  • n/2^n*z²n
  • n/2 to the power of n*z to the power of 2n
  • n/2^nz^2n
  • n/2nz2n
  • n divide by 2^n*z^2n
Sum of series/ n/2^n*z^2n

Sum of series n/2^n*z^2n


∑
=

The solution

You have entered [src] 
  oo         
____         
\   `        
 \    n   2  
  \   --*z *n
  /    n     
 /    2      
/___,        
n = 1
$$\sum_{n=1}^{\infty} n z^{2} \frac{n}{2^{n}}$$ 
Sum(((n/2^n)*z^2)*n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n z^{2} \frac{n}{2^{n}}$$
It is a series of species
$$a_{n} \left(c z - z_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{z_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = n^{2} z^{2}$$
and
$$z_{0} = -2$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{n^{2}}{\left(n + 1\right)^{2}}\right)\right)$$
Let's take the limit
we find
False

$$R = 0$$
The answer [src] 
   2
6*z
$$6 z^{2}$$ 
6*z^2

    Examples of finding the sum of a series

    © Mister Exam – Calculator