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(3-sinn)/(n-lnn)
Sum of series/ (3-sinn)/(n-lnn)

Sum of series (3-sinn)/(n-lnn)


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The solution

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  oo            
 ___            
 \  `           
  \   3 - sin(n)
   )  ----------
  /   n - log(n)
 /__,           
n = 1
$$\sum_{n=1}^{\infty} \frac{3 - \sin{\left(n \right)}}{n - \log{\left(n \right)}}$$ 
Sum((3 - sin(n))/(n - log(n)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3 - \sin{\left(n \right)}}{n - \log{\left(n \right)}}$$
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
$$a_{n} = \frac{3 - \sin{\left(n \right)}}{n - \log{\left(n \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(\sin{\left(n \right)} - 3\right) \left(n - \log{\left(n + 1 \right)} + 1\right)}{\left(n - \log{\left(n \right)}\right) \left(\sin{\left(n + 1 \right)} - 3\right)}}\right|$$
Let's take the limit
we find
$$1 = \left|{\left\langle -2, - \frac{1}{2}\right\rangle}\right|$$
False
The rate of convergence of the power series
The graph
Sum of series (3-sinn)/(n-lnn)

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