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

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


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The solution

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

False

False
The rate of convergence of the power series
Numerical answer [src] 
0.104556251724813703737068273536
0.104556251724813703737068273536
The graph
Sum of series (1-1/n)^(n^2)

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