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1/2sqrt(2)+1/3sqrt(3)+1/(n+1)sqrt(n+1)
Sum of series/ 1/2sqrt(2)+1/3sqrt(3)+1/(n+1)sqrt(n+1)

Sum of series 1/2sqrt(2)+1/3sqrt(3)+1/(n+1)sqrt(n+1)


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The solution

You have entered [src] 
  oo                             
____                             
\   `                            
 \    /  ___     ___     _______\
  \   |\/ 2    \/ 3    \/ n + 1 |
  /   |----- + ----- + ---------|
 /    \  2       3       n + 1  /
/___,                            
n = 1
$$\sum_{n=1}^{\infty} \left(\frac{\sqrt{n + 1}}{n + 1} + \left(\frac{\sqrt{3}}{3} + \frac{\sqrt{2}}{2}\right)\right)$$ 
Sum(sqrt(2)/2 + sqrt(3)/3 + sqrt(n + 1)/(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sqrt{n + 1}}{n + 1} + \left(\frac{\sqrt{3}}{3} + \frac{\sqrt{2}}{2}\right)$$
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
$$a_{n} = \frac{\sqrt{3}}{3} + \frac{\sqrt{2}}{2} + \frac{1}{\sqrt{n + 1}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\frac{\sqrt{3}}{3} + \frac{\sqrt{2}}{2} + \frac{1}{\sqrt{n + 1}}}{\frac{\sqrt{3}}{3} + \frac{\sqrt{2}}{2} + \frac{1}{\sqrt{n + 2}}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo                             
____                             
\   `                            
 \    /              ___     ___\
  \   |    1       \/ 2    \/ 3 |
   )  |--------- + ----- + -----|
  /   |  _______     2       3  |
 /    \\/ 1 + n                 /
/___,                            
n = 1
$$\sum_{n=1}^{\infty} \left(\frac{\sqrt{3}}{3} + \frac{\sqrt{2}}{2} + \frac{1}{\sqrt{n + 1}}\right)$$ 
Sum(1/sqrt(1 + n) + sqrt(2)/2 + sqrt(3)/3, (n, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series 1/2sqrt(2)+1/3sqrt(3)+1/(n+1)sqrt(n+1)

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