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(5/2^n+1)+ln(n/n+1)
Sum of series/ (5/2^n+1)+ln(n/n+1)

Sum of series (5/2^n+1)+ln(n/n+1)


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The solution

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  oo                         
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  \   /   n          /n    \\
   )  |5/2  + 1 + log|- + 1||
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n = 1
$$\sum_{n=1}^{\infty} \left(\left(\left(\frac{5}{2}\right)^{n} + 1\right) + \log{\left(1 + \frac{n}{n} \right)}\right)$$ 
Sum((5/2)^n + 1 + log(n/n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\left(\frac{5}{2}\right)^{n} + 1\right) + \log{\left(1 + \frac{n}{n} \right)}$$
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
$$a_{n} = \left(\frac{5}{2}\right)^{n} + \log{\left(2 \right)} + 1$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\frac{5}{2}\right)^{n} + \log{\left(2 \right)} + 1}{\left(\frac{5}{2}\right)^{n + 1} + \log{\left(2 \right)} + 1}\right)$$
Let's take the limit
we find
False

False

False
The rate of convergence of the power series
The answer [src] 
oo
$$\infty$$ 
oo
Numerical answer
The series diverges
The graph
Sum of series (5/2^n+1)+ln(n/n+1)

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