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logn/n^(5/4)

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  • Sum of series:
  • 2^n/n^2 2^n/n^2
  • sqrt(n/n^3+2*n+9) sqrt(n/n^3+2*n+9)
  • sen(1/n) sen(1/n)
  • tan(n*x)

  • Identical expressions

  • logn/n^(five / four)
  • logarithm of n divide by n to the power of (5 divide by 4)
  • logarithm of n divide by n to the power of (five divide by four)
  • logn/n(5/4)
  • logn/n5/4
  • logn/n^5/4
  • logn divide by n^(5 divide by 4)
Sum of series/ logn/n^(5/4)

Sum of series logn/n^(5/4)


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The solution

You have entered [src] 
  oo        
____        
\   `       
 \    log(n)
  \   ------
  /     5/4 
 /     n    
/___,       
n = 1
∑n=1∞log⁡(n)n54\sum_{n=1}^{\infty} \frac{\log{\left(n \right)}}{n^{\frac{5}{4}}} 
Sum(log(n)/n^(5/4), (n, 1, oo))
The radius of convergence of the power series
Given number:
log⁡(n)n54\frac{\log{\left(n \right)}}{n^{\frac{5}{4}}}
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)n54a_{n} = \frac{\log{\left(n \right)}}{n^{\frac{5}{4}}}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞((n+1)54∣log⁡(n)∣n54log⁡(n+1))1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{\frac{5}{4}} \left|{\log{\left(n \right)}}\right|}{n^{\frac{5}{4}} \log{\left(n + 1 \right)}}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.502
The graph
Sum of series logn/n^(5/4)

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