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Sum of series/ x^n/sqrt(n+1)

Sum of series x^n/sqrt(n+1)


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

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