Mister Exam

# Sum of series ln(n/(n+1))

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### The solution

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\      /  n  \
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/      \n + 1/
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n = 1           
$$\sum_{n=1}^{\infty} \log{\left(\frac{n}{n + 1} \right)}$$
Sum(log(n/(n + 1)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\log{\left(\frac{n}{n + 1} \right)}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \log{\left(\frac{n}{n + 1} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\log{\left(\frac{n}{n + 1} \right)}}{\log{\left(\frac{n + 1}{n + 2} \right)}}\right)$$
Let's take the limit
we find
True

False`
The rate of convergence of the power series
The graph
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