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

Answer 1

#R=1/e#

Use the ratio test, which states that #suma_n# converges if #L<1#, where #L=lim_(nrarroo)abs(a_(n+1)/a_n)#. Here,
#L=lim_(nrarroo)abs((n+1)^(n+1)/((n+1)!)x^(n+1)(n!)/n^n1/x^n)#

Streamlining:

#L=lim_(nrarroo)abs((n+1)^(n+1)/n^n((n!)/((n+1)n!))x^(n+1)/x^n)#
#L=lim_(nrarroo)abs((n+1)^(n+1)/(n+1)1/n^nx)#
Bringing the #x# out of the limit, since the limit depends only on how #n# changes:
#L=absxlim_(nrarroo)abs((n+1)^n/n^n)#
#L=absxlim_(nrarroo)abs(((n+1)/n)^n)#
This is a well-known limit that approaches #e#:
#L=eabsx#
The posted series will converge when #L<1#, so the interval of convergence will be on when:
#eabsx<1#
#absx<1/e#
Thus the radius of convergence is #R=1/e#, centered at #x=0#.
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Answer 2
To find the radius of convergence for the series \(\sum_{n=1}^{\infty} \frac{n^n}{n!} x^n\), we can use the ratio test. The ratio test states that if \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L\), then the series converges if \(L < 1\), diverges if \(L > 1\), and the test is inconclusive if \(L = 1\). For our series, \(a_n = \frac{n^n}{n!} x^n\). Therefore, \[ \begin{align*} \left| \frac{a_{n+1}}{a_n} \right| &= \left| \frac{(n+1)^{n+1}}{(n+1)!} \cdot \frac{n!}{n^n} \cdot x^{n+1} \cdot \frac{1}{x^n} \right| \\ &= \left| \frac{(n+1)^{n+1}}{(n+1)n^n} \cdot x \right| \\ &= \left| \frac{(n+1)^n}{n^n} \cdot \frac{(n+1)}{n+1} \cdot x \right| \\ &= \left| \left(1 + \frac{1}{n}\right)^n \cdot x \right| \end{align*} \] As \(n\) approaches infinity, \(\left(1 + \frac{1}{n}\right)^n\) approaches \(e\), where \(e\) is the base of the natural logarithm. Therefore, \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = |e \cdot x| \] The series converges if \(|e \cdot x| < 1\). Thus, the radius of convergence is \(R = \frac{1}{e}\), where \(e\) is approximately \(2.71828\). Therefore, the radius of convergence is approximately \(R \approx 0.3679\).
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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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