Calculate whether the following sequence diverges or converges.

Asked Jun 03 '14 at 14:27

Active Jun 03 '14 at 14:54

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Could someone please confirm my steps below, I am not sure if I have done this right. $$ \lim_{n \to \infty} \frac{n^n}{(n+3)^{n+1}} $$

For large values of n: $$ \lim_{n \to \infty} \frac{n^n}{(n+3)^{n+1}} = \lim_{n \to \infty} \frac{n^n}{n^{n+1}} =\lim_{n \to \infty} \frac{1}{n} = 0 $$ Hence the sequence converges. Is this correct and if so, is there a better way to show this? (This is for an exam so I am not sure if this is enough working out for this question)

asked Jun 03 '14 at 14:27

baddin

3

Write the terms as ${1\over n+3}{1\over\Bigl(1+{3\over n}\Bigr)^{n}}$. – David Mitra Jun 03 '14 at 14:30
Ahh perfect, exactly what I was looking, Thanks very much!!! – baddin Jun 03 '14 at 14:33
1

Actually, your solution is better; but use the Squeeze Theorem. ($0\le {n^n\over (n+3)^{n+1}}\le{1\over n}$.) – David Mitra Jun 03 '14 at 14:34
Good idea, so is it fine to say n^n/(4n)^(n+1)< The sequence < 1/n (OOps just saw your result) – baddin Jun 03 '14 at 14:38

Calculate whether the following sequence diverges or converges.

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