Mister Exam

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

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The solution

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

False
The rate of convergence of the power series
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____
\   
\      ___   ___
\   \/ 2 *\/ n
)  -----------
/      _______
/     \/ 1 + n
/___,
n = 0            
$$\sum_{n=0}^{\infty} \frac{\sqrt{2} \sqrt{n}}{\sqrt{n + 1}}$$
Sum(sqrt(2)*sqrt(n)/sqrt(1 + n), (n, 0, oo))
The graph
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