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

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


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
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  \    -1 - n  n
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n = 1
$$\sum_{n=1}^{\infty} 2^{- n - 1} n^{n}$$ 
Sum(2^(-1 - n)*n^n, (n, 1, oo))
The graph
Sum of series n^n/2^(n+1)

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