Showing a sequence is increasing (without derivative)

Question

I have looked through several similar post but can't find one that quite fits my problem. The sequence in question is

$$f(n) = \left(1+\frac{1}{n}\right)^n$$

which obviously can't just be derived. I have been trying to prove this by induction; however the algebra doesn't give a nice form that makes it clear that $f(n+1)>f(n)$. Any help or insight is appreciated!

Note that $f(n)=\left(\frac{n+1}{n}\right)^n$. Try dividing $f(n+1)$ by $f(n)$ — saulspatz, Sep 11 '20 at 20:56
See [THIS ANSWER}(https://math.stackexchange.com/questions/1589429/how-to-prove-that-logxx-when-x1/1590263#1590263) in which I showed that $\left(1+\frac xn\right)^n$ is decreasing for $x>-n$. — Mark Viola, Sep 11 '20 at 21:18

score 0 · Answer 1 · answered Sep 11 '20 at 21:24

As indicated in the comments we have that

$$\frac{f(n+1)}{f(n)}=\frac{\left(\frac{n+2}{n+1}\right)^{n+1}}{\left(\frac{n+1}{n}\right)^n}=\left(\frac{n+2}{n+1}\right)\left(\frac{n(n+2)}{(n+1)^2}\right)^n=$$

$$=\left(\frac{n+2}{n+1}\right)\left(\frac{(n+1)^2-1}{(n+1)^2}\right)^n=\left(\frac{n+2}{n+1}\right)\left(1-\frac{1}{(n+1)^2}\right)^n$$

then use Bernoulli's inequality.

Showing a sequence is increasing (without derivative)

1 Answers1