Estimate (1+a)^n from above.

Question

I am doing some estimations and I cannot think out inequality which should help.

I have $(1+a)^n$ where $a$ is constant very close to $0$. I want to have something of the following form $(1+a)^n\leq1+k(a,n)$.

I found somewhere $(1+a)^n\leq1+\frac{na}{1-na}$ but I have doubts that it is correct.

Answer 1

Since $e^x\geq 1+x$ by convexity, $$ (1+a)^n \leq \exp(an)=\frac{1}{\exp(-an)} \leq \frac{1}{1-an}.$$