10.3, #11. We multiple the two series:

e^{t} cos t = (1 + t + \frac{1}{2}t^{2} + \frac{1}{6}t^{3} + ⋯)(1 − \frac{1}{2}t^{2} + \frac{1}{24}t^{4} + ⋯)

= 1 + t + (\frac{1}{2}t^{2} − \frac{1}{2}t^{2}) + (\frac{1}{6}t^{3} − \frac{1}{2}t^{3}) + (\frac{1}{24}t^{4} − \frac{1}{4}t^{4} + \frac{1}{24}t^{4}) + ⋯

= 1 + t − \frac{1}{3}t^{3} − \frac{1}{6}t^{4} + · · · .

10.3, #15. We expand the three functions in series about y = 0:

ln(1 + y^{2}) = y^{2} − \frac{1}{2}y^{4} + ⋯
sin(y^{2}) = y^{2} − \frac{1}{6}y^{6} + ⋯
1 − cos y = \frac{1}{2}y^{2} − \frac{1}{24}y^{4} + · · · .

So for y near 0, but y ≠ 0, we have

sin(y^{2}) > ln(1 + y^{2}) > 1 − cos y > 0.

10.3, #16. We expand the three functions in series about θ = 0:

\begin{aligned}1 + sin θ & = 1 + θ − \frac{1}{6}θ^{3} + ⋯ \\ cos θ & = 1 − \frac{1}{2}θ^{2} + ⋯ \\ \frac{1}{1 − θ^{2}} & = 1 + θ^{2} + · · · . \end{aligned}

So for θ > 0, we have

cos θ < \frac{1}{1 − θ^{2}} < 1 + sin θ.

10.3, #21.

E = \frac{kQ}{(R − 1)^{2}} − \frac{kQ}{(R + 1)^{2}} = \frac{kQ}{R^{2}}(\frac{1}{(1 − 1/R)^{2}} − \frac{1}{(1 + 1/R)^{2}} )

= \frac{kQ}{R^{2}}(1 + 2\frac{1}{R} + 3\frac{1}{R^{3}} + ⋯ − 1 + 2\frac{1}{R} − 3\frac{1}{R^{3}} + ⋯)

= \frac{4kQ}{R^{3}}(1 + \frac{2}{R^{2}} + \frac{3}{R^{4}} + ⋯)

10.3, #23.

V = 2πσ(\sqrt{R^{2} + a^{2}} − R)

= 2πσR(\sqrt{1 + (a/R)^{2}} − 1)

= 2πσR(1 + \frac{1}{2}(a/R)^{2} + ⋯ − 1)

= \frac{πσa^{2}}{R} + higher order terms.

10.3, #26.

F = \frac{mgR^{2}}{(r + h)^{2}} = mg \frac{1}{(1 + h/R)^{2}}
= mg(1 − 2(h/r) + 3(h/R)^{2} − + ⋯)
= mg