Interesting limit

Question

Let $a_n$ a sequence of real numbers such that $\lim_{n\to +\infty}a_n=A$. Is it true that

$$\lim_{n\to +\infty}\Big(1+\frac{a_n}{n}\Big)^n=e^A?$$

Yes. It is an easy consequence of the fact that $(1+A/n)^n\to e^A$. — Paramanand Singh, Oct 30 '18 at 14:52

score 2 · Answer 1 · answered Oct 30 '18 at 14:50

Assuming $\;A\;$ is finite, use arithmetic of limits and the basic fact that

$$c_n\xrightarrow[n\to\infty]{}\infty \implies\left(1+\frac1{c_n}\right)^{c_n}\xrightarrow[n\to\infty]{}e$$

to deduce that

$$\left(1+\frac{a_n}n\right)^n=\left[\left(1+\frac1{\frac n{a_n}}\right)^{\frac n{a_n}}\right]^{a_n}\xrightarrow[n\to\infty]{}e^A$$

Can you see what happens if $\;\lim a_n=\pm\infty\;$ ?

1 Answers1