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Sum of series/ log(1-3/n)

Sum of series log(1-3/n)


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The solution

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n = 1
∑n=1∞log⁡(1−3n)\sum_{n=1}^{\infty} \log{\left(1 - \frac{3}{n} \right)} 
Sum(log(1 - 3/n), (n, 1, oo))
The radius of convergence of the power series
Given number:
log⁡(1−3n)\log{\left(1 - \frac{3}{n} \right)}
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⁡(1−3n)a_{n} = \log{\left(1 - \frac{3}{n} \right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞∣log⁡(1−3n)log⁡(1−3n+1)∣1 = \lim_{n \to \infty} \left|{\frac{\log{\left(1 - \frac{3}{n} \right)}}{\log{\left(1 - \frac{3}{n + 1} \right)}}}\right|
Let's take the limit
we find
True

False
Numerical answer [src] 
Sum(log(1 - 3/n), (n, 1, oo))
Sum(log(1 - 3/n), (n, 1, oo))

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