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(-1)^(n+1)/log(n+2^0.5)

Sum of series (-1)^(n+1)/log(n+2^0.5)



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The solution

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  oo                
____                
\   `               
 \          n + 1   
  \     (-1)        
   )  --------------
  /      /      ___\
 /    log\n + \/ 2 /
/___,               
n = 1               
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{\log{\left(n + \sqrt{2} \right)}}$$
Sum((-1)^(n + 1)/log(n + sqrt(2)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n + 1}}{\log{\left(n + \sqrt{2} \right)}}$$
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
$$a_{n} = \frac{\left(-1\right)^{n + 1}}{\log{\left(n + \sqrt{2} \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\log{\left(n + 1 + \sqrt{2} \right)}}{\log{\left(n + \sqrt{2} \right)}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The graph
Sum of series (-1)^(n+1)/log(n+2^0.5)

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