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abs(e^(cos(n)/n)-cos1/n)
Sum of series/ abs(e^(cos(n)/n)-cos1/n)

Sum of series abs(e^(cos(n)/n)-cos1/n)


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The solution

You have entered [src] 
  oo                    
____                    
\   `                   
 \    | cos(n)         |
  \   | ------         |
   )  |   n      cos(1)|
  /   |E       - ------|
 /    |            n   |
/___,                   
n = 1
$$\sum_{n=1}^{\infty} \left|{e^{\frac{\cos{\left(n \right)}}{n}} - \frac{\cos{\left(1 \right)}}{n}}\right|$$ 
Sum(Abs(E^(cos(n)/n) - cos(1)/n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left|{e^{\frac{\cos{\left(n \right)}}{n}} - \frac{\cos{\left(1 \right)}}{n}}\right|$$
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
$$a_{n} = \left|{e^{\frac{\cos{\left(n \right)}}{n}} - \frac{\cos{\left(1 \right)}}{n}}\right|$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left|{e^{\frac{\cos{\left(n \right)}}{n}} - \frac{\cos{\left(1 \right)}}{n}}\right|}{\left|{e^{\frac{\cos{\left(n + 1 \right)}}{n + 1}} - \frac{\cos{\left(1 \right)}}{n + 1}}\right|}}\right|$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo                      
____                      
\   `                     
 \    |            cos(n)|
  \   |            ------|
   )  |  cos(1)      n   |
  /   |- ------ + e      |
 /    |    n             |
/___,                     
n = 1
$$\sum_{n=1}^{\infty} \left|{e^{\frac{\cos{\left(n \right)}}{n}} - \frac{\cos{\left(1 \right)}}{n}}\right|$$ 
Sum(Abs(-cos(1)/n + exp(cos(n)/n)), (n, 1, oo))
The graph
Sum of series abs(e^(cos(n)/n)-cos1/n)

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