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3-sin*n/n-ln*n
Sum of series/ 3-sin*n/n-ln*n

Sum of series 3-sin*n/n-ln*n


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The solution

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

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