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 \[\int_{1}^{x} \frac{dt}{t^{1-(1/n)}} – \int_{1}^{x} \frac{dt}{t^{1+(1/n)}} < \frac{1}{n} \int_{1}^{x} \left(t – \frac{1}{t}\right) \frac{dt}{t} = \frac{1}{n} \left(x + \frac{1}{x} – 2\right).\] Hence deduce the results of § 209.
3. If \(\xi_{n}\) is a function of \(n\) such that \(n\xi_{n} \to l\) as \(n \to \infty\), 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 \(n\xi_{n} \to \infty\), then \((1 + \xi_{n})^{n} \to \infty\); and if \(1 + \xi_{n} > 0\) and \(n\xi_{n} \to -\infty\), 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 \(y\).
