212. Series connected with the exponential and logarithmic functions. Expansion of $e^x$ by Taylor’s Theorem.

1. Show that $$\cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\dots ,\sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\dots .$$

2. If $x$ is positive then the greatest term in the exponential series is the $([x]+1)$-th, unless $x$ is an integer, when the preceding term is equal to it.

3. Show that $n!>(n/e{)}^n$. [For $n^n/n!$ is one term in the series for $e^n$.]

4. Prove that $e^n=(n^n/n!)(2+S_1+S_2)$, where $$S_1=\frac{1}{1+\nu }+\frac{1}{(1+\nu )(1+2\nu )}+\dots ,S_2=(1–\nu )+(1–\nu )(1–2\nu )+\dots ,$$ and $\nu =1/n$; and deduce that $n!$ lies between $2(n/e{)}^n$ and $2(n+1)(n/e{)}^n$.

5. Employ the exponential series to prove that $e^x$ tends to infinity more rapidly than any power of $x$. [Use the inequality $e^x>x^n/n!$.]

6. Show that $e$ is not a rational number. [If $e=p/q$, where $p$ and $q$ are integers, we must have $$\frac{p}{q}=1+1+\frac{1}{2!}+\frac{1}{3!}+\cdots +\frac{1}{q!}+\dots$$ or, multiplying up by $q!$, $$q!(\frac{p}{q}–1–1–\frac{1}{2!}–\cdots –\frac{1}{q!})=\frac{1}{q+1}+\frac{1}{(q+1)(q+2)}+\dots$$ and this is absurd, since the left-hand side is integral, and the right-hand side less than $\{1/(q+1)\}+\{1/(q+1){\}}^2+\cdots =1/q$.]

7. Sum the series $\sum _0^{\mathrm{\infty }}{P}_r(n){\displaystyle \frac{x^n}{n!}}$, where $P_r(n)$ is a polynomial of degree $r$ in $n$. [We can express $P_r(n)$ in the form $$A_0+A_{1}n+A_{2}n(n–1)+\cdots +A_{r}n(n–1)\dots (n–r+1),$$ and $$\begin{array}{rl}\sum _0^{\mathrm{\infty }}{P}_r(n)\frac{x^n}{n!}& =A_0\sum _0^{\mathrm{\infty }}\frac{x^n}{n!}+A_1\sum _1^{\mathrm{\infty }}\frac{x^n}{(n–1)!}+\cdots +A_r\sum _r^{\mathrm{\infty }}\frac{x^n}{(n–r)!}\\ & =(A_0+A_{1}x+A_2{x}^2+\cdots +A_r{x}^r)e^x.]\end{array}$$

8. Show that $$\sum _1^{\mathrm{\infty }}\frac{n^3}{n!}{x}^n=(x+3x^2+x^3)e^x,\sum _1^{\mathrm{\infty }}\frac{n^4}{n!}{x}^n=(x+7x^2+6x^3+x^4)e^x;$$ and that if $S_n=1^3+2^3+\cdots +n^3$ then $$\sum _1^{\mathrm{\infty }}{S}_n\frac{x^n}{n!}=\frac{1}{4}(4x+14x^2+8x^3+x^4)e^x.$$ In particular the last series is equal to zero when $x=-2$.

9. Prove that $\sum (n/n!)=e$, $\sum (n^2/n!)=2e$, $\sum (n^3/n!)=5e$, and that $\sum (n^k/n!)$, where $k$ is any positive integer, is a positive integral multiple of $e$.

10. Prove that $\sum _1^{\mathrm{\infty }}{\displaystyle \frac{(n–1)x^n}{(n+2)n!}}=\{(x^2–3x+3)e^x+\frac{1}{2}{x}^2–3\}/x^2$.

[Multiply numerator and denominator by $n+1$, and proceed as in Ex. 7.]

11. Determine $a$, $b$, $c$ so that $\{(x+a)e^x+(bx+c)\}/x^3$ tends to a limit as $x\to 0$, evaluate the limit, and draw the graph of the function $e^x+{\displaystyle \frac{bx+c}{x+a}}$.

12. Draw the graphs of $1+x$, $1+x+\frac{1}{2}{x}^2$, $1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3$, and compare them with that of $e^x$.

13. Prove that $e^{-x}–1+x–{\displaystyle \frac{x^n}{2!}}+\cdots –(-1{)}^n{\displaystyle \frac{x^n}{n!}}$ is positive or negative according as $n$ is odd or even. Deduce the exponential theorem.

14. If $$X_0=e^x,X_1=e^x–1,X_2=e^x–1–x,X_3=e^x–1–x–(x^2/2!),\dots ,$$ then $d{X}_{\nu }/dx=X_{\nu -1}$. Hence prove that if $t>0$ then $$X_1(t)={\int }_0^t{X}_{0}dx<t{e}^t,X_2(t)={\int }_0^t{X}_{1}dx<{\int }_0^{t}x{e}^{x}dx<e^t{\int }_0^{t}xdx=\frac{t^2}{2!}{e}^t,$$ and generally $X_{\nu }(t)<{\displaystyle \frac{t^{\nu }}{\nu !}}{e}^t$. Deduce the exponential theorem.

15. Show that the expansion in powers of $p$ of the positive root of $x^{2+p}=a^2$ begins with the terms $$a\{1–\frac{1}{2}p\log a+\frac{1}{8}{p}^2\log a(2+\log a)\}.$$