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ln(sqrt(n))/4^n

  • How to use it?

  • Sum of series:
  • 1/n^4 1/n^4
  • yi
  • 1/(n*log(n)) 1/(n*log(n))
  • ln(sqrt(n))/4^n ln(sqrt(n))/4^n

  • Identical expressions

  • ln(sqrt(n))/ four ^n
  • ln( square root of (n)) divide by 4 to the power of n
  • ln( square root of (n)) divide by four to the power of n
  • ln(√(n))/4^n
  • ln(sqrt(n))/4n
  • lnsqrtn/4n
  • lnsqrtn/4^n
  • ln(sqrt(n)) divide by 4^n
Sum of series/ ln(sqrt(n))/4^n

Sum of series ln(sqrt(n))/4^n


∑
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The solution

You have entered [src] 
  oo            
____            
\   `           
 \       /  ___\
  \   log\\/ n /
   )  ----------
  /        n    
 /        4     
/___,           
n = 1
∑n=1∞log⁡(n)4n\sum_{n=1}^{\infty} \frac{\log{\left(\sqrt{n} \right)}}{4^{n}} 
Sum(log(sqrt(n))/4^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
log⁡(n)4n\frac{\log{\left(\sqrt{n} \right)}}{4^{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=log⁡(n)a_{n} = \log{\left(\sqrt{n} \right)}
and
x0=−4x_{0} = -4
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−4+lim⁡n→∞∣log⁡(n)log⁡(n+1)∣)\frac{1}{R} = \tilde{\infty} \left(-4 + \lim_{n \to \infty} \left|{\frac{\log{\left(\sqrt{n} \right)}}{\log{\left(\sqrt{n + 1} \right)}}}\right|\right)
Let's take the limit
we find
False

R=0R = 0
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.000.05
The answer [src] 
  oo                
 ___                
 \  `               
  \    -n    /  ___\
  /   4  *log\\/ n /
 /__,               
n = 1
∑n=1∞4−nlog⁡(n)\sum_{n=1}^{\infty} 4^{- n} \log{\left(\sqrt{n} \right)} 
Sum(4^(-n)*log(sqrt(n)), (n, 1, oo))
Numerical answer [src] 
0.0340368668699901685928816555060
0.0340368668699901685928816555060
The graph
Sum of series ln(sqrt(n))/4^n

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