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ln(n/10)/(exp(n/10)-1)

Sum of series ln(n/10)/(exp(n/10)-1)



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The solution

You have entered [src]
   oo          
______         
\     `        
 \         /n \
  \     log|--|
   \       \10/
    \   -------
    /    n     
   /     --    
  /      10    
 /      e   - 1
/_____,        
 n = 1         
n=1log(n10)en101\sum_{n=1}^{\infty} \frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} - 1}
Sum(log(n/10)/(exp(n/10) - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
log(n10)en101\frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} - 1}
It is a series of species
an(cxx0)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+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=log(n10)en101a_{n} = \frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} - 1}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn(en10+1101)log(n10)(en101)log(n10+110)1 = \lim_{n \to \infty} \left|{\frac{\left(e^{\frac{n}{10} + \frac{1}{10}} - 1\right) \log{\left(\frac{n}{10} \right)}}{\left(e^{\frac{n}{10}} - 1\right) \log{\left(\frac{n}{10} + \frac{1}{10} \right)}}}\right|
Let's take the limit
we find
False

False

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.5-40-20
Numerical answer [src]
-34.2737296704091840084709480062
-34.2737296704091840084709480062
The graph
Sum of series ln(n/10)/(exp(n/10)-1)

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