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1/2^nn
Sum of series/ 1/2^nn

Sum of series 1/2^nn


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The solution

You have entered [src] 
  oo       
 ___       
 \  `      
  \    -n  
  /   2  *n
 /__,      
n = 1
∑n=1∞(12)nn\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^{n} n 
Sum((1/2)^n*n, (n, 1, oo))
The radius of convergence of the power series
Given number:
(12)nn\left(\frac{1}{2}\right)^{n} n
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=na_{n} = n
and
x0=−2x_{0} = -2
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−2+lim⁡n→∞(nn+1))\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{n}{n + 1}\right)\right)
Let's take the limit
we find
False

R=0R = 0
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.502
The answer [src] 
2
22 
2
Numerical answer [src] 
2.00000000000000000000000000000
2.00000000000000000000000000000
The graph
Sum of series 1/2^nn

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