209. The representation of $\log x$ as a limit

1. Prove, by taking $y=1$ and $n=6$ in the inequalities (4) of § 208, that $2.5<e<2.9$.

2. Prove that if $t>1$ then $(t^{1/n}–t^{-1/n})/(t–t^{-1})<1/n$, and so that if $x>1$ then $${\int }_1^x\frac{dt}{t^{1-(1/n)}}–{\int }_1^x\frac{dt}{t^{1+(1/n)}}<\frac{1}{n}{\int }_1^x\left(t–\frac{1}{t}\right)\frac{dt}{t}=\frac{1}{n}(x+\frac{1}{x}–2).$$ Hence deduce the results of § 209.

3. If ${\xi }_n$ is a function of $n$ such that $n{\xi }_n\to l$ as $n\to \mathrm{\infty }$, then $(1+{\xi }_n{)}^n\to e^l$. [Writing $n\log (1+{\xi }_n)$ in the form $$l\left(\frac{n{\xi }_n}{l}\right)\frac{\log (1+{\xi }_n)}{{\xi }_n},$$ and using Ex. LXXXII. 4, we see that $n\log (1+{\xi }_n)\to l$.]

4. If $n{\xi }_n\to \mathrm{\infty }$, then $(1+{\xi }_n{)}^n\to \mathrm{\infty }$; and if $1+{\xi }_n>0$ and $n{\xi }_n\to -\mathrm{\infty }$, then $$(1+{\xi }_n{)}^n\to 0.$$

5. Deduce from (1) of § 208 the theorem that $e^y$ tends to infinity more rapidly than any power of $y$.