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

Sum of series ln(sqrtn+1/n)


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The solution

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  oo                
 ___                
 \  `               
  \      /  ___   1\
   )  log|\/ n  + -|
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n = 4
$$\sum_{n=4}^{\infty} \log{\left(\sqrt{n} + \frac{1}{n} \right)}$$ 
Sum(log(sqrt(n) + 1/n), (n, 4, oo))
The radius of convergence of the power series
Given number:
$$\log{\left(\sqrt{n} + \frac{1}{n} \right)}$$
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
$$a_{n} = \log{\left(\sqrt{n} + \frac{1}{n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\log{\left(\sqrt{n} + \frac{1}{n} \right)}}{\log{\left(\sqrt{n + 1} + \frac{1}{n + 1} \right)}}}\right|$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The graph
Sum of series ln(sqrtn+1/n)

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