$lim_{j->\infty} (j^j)/((j+1)^{j})$

Question

Can someone please explain this limit:

$$lim_{j\rightarrow\infty} \frac{j^{j}}{(j+1)^{j}}=\frac{1}{e}?$$

I got it from this series:

$$\sum_1^{\infty}\frac{j!}{j^j}.$$

score 2 · Accepted Answer · answered Apr 27 '16 at 13:10

2

The reciprocal is $\left(1+\frac1j\right)^j$, which is one of the definitions of $e$.

answered Apr 27 '16 at 13:10

Empy2

I know this limit but I don't see how to get there from where I am. – Strange Brew Apr 27 '16 at 13:11
2

divide top and bottom by $j^j$ to get $\frac{j^j}{(j+1)^j}=\frac{1}{(1+1/j)^j}$ – Conrad Turner Apr 27 '16 at 13:13
Ah, I see! Thanks! – Strange Brew Apr 27 '16 at 13:15

score 0 · Answer 2 · answered Apr 27 '16 at 13:12

0

$$L=\lim_{j\rightarrow\infty} \frac{j^{j}}{(j+1)^{j}}=\lim_{j\rightarrow\infty} \frac{1}{(1+\frac1j)^{j}}$$

The limit is of the form $1^{\infty}$.

Thus, $$L=e^{\lim_{j\rightarrow\infty}\left(\frac{1}{1+\frac1j}-1\right)j}=e^{-1}$$

answered Apr 27 '16 at 13:12

GoodDeeds

2 Answers2