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sqrt(n+1)/2^n
Sum of series/ sqrt(n+1)/2^n

Sum of series sqrt(n+1)/2^n


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The solution

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n = 1
∑n=1∞n+12n\sum_{n=1}^{\infty} \frac{\sqrt{n + 1}}{2^{n}} 
Sum(sqrt(n + 1)/2^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
n+12n\frac{\sqrt{n + 1}}{2^{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=n+1a_{n} = \sqrt{n + 1}
and
x0=−2x_{0} = -2
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−2+lim⁡n→∞(n+1n+2))\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{\sqrt{n + 1}}{\sqrt{n + 2}}\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] 
  oo               
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  \    -n   _______
  /   2  *\/ 1 + n 
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n = 1
∑n=1∞2−nn+1\sum_{n=1}^{\infty} 2^{- n} \sqrt{n + 1} 
Sum(2^(-n)*sqrt(1 + n), (n, 1, oo))
Numerical answer [src] 
1.69450750547150138439924692807
1.69450750547150138439924692807
The graph
Sum of series sqrt(n+1)/2^n

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