How do you find the radius of convergence #Sigma (n!)/n^n x^n# from #n=[1,oo)#?

Answer 1

The radius of convergence is #R = e#

Applying the test of ratios:

#abs (a_(n+1)/a_n) = abs ( ( ( (n+1)! x^(n+1))/(n+1)^(n+1)) /( ( n!x^n)/n^n))#
#abs (a_(n+1)/a_n) = ((n+1)!)/(n!) xx n^n/(n+1)^(n+1) xx abs (x^(n+1)/x^n)#
#abs (a_(n+1)/a_n) = (n+1) xx 1/(n+1) (n/(n+1))^n xx abs (x)#
#abs (a_(n+1)/a_n) =abs (x)/((n+1)/n)^n #
#abs (a_(n+1)/a_n) =abs (x)/(1+1/n)^n #

So:

#lim_(n->oo) abs (a_(n+1)/a_n) = absx lim_(n->oo) 1/(1+1/n)^n = absx/e#
So the radius of convergence is #R = e# which means that the series is absolutely convergent for #absx < e#
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Answer 2

To find the radius of convergence of the series ( \sum_{n=1}^{\infty} \frac{n!}{n^n} x^n ), you can use the ratio test.

The ratio test states that for a series ( \sum_{n=1}^{\infty} a_n x^n ), the radius of convergence ( R ) is given by:

[ R = \lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right| ]

Here, ( a_n = \frac{n!}{n^n} ).

[ R = \lim_{n \to \infty} \left| \frac{\frac{n!}{n^n}}{\frac{(n+1)!}{(n+1)^{n+1}}} \right| ]

[ R = \lim_{n \to \infty} \left| \frac{n!}{n^n} \cdot \frac{(n+1)^n}{(n+1)!} \right| ]

[ R = \lim_{n \to \infty} \left| \frac{(n+1)^n}{n^n} \right| ]

[ R = \lim_{n \to \infty} \left| \left(1 + \frac{1}{n}\right)^n \right| ]

The limit on the right-hand side is known to converge to ( e ), Euler's number.

So, the radius of convergence ( R ) for the given series is ( R = e ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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