1. Estimate oredr O(h)

Consider non-dimensional differential equation for the height at the highest point is given by
$$h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} log_e(1+\mu)$$
$0<\mu<<1.$
Deduce an estimate to $O(\mu)$ for $h(\mu)$ and compare with $t_h(\mu)=1-\frac{\mu}{2}+...$

2. Re: Estimate oredr O(h)

I would start with a Taylor's series expansion of $\displaystyle ln(1+ \mu)$.

3. Re: Estimate oredr O(h)

Originally Posted by HallsofIvy
I would start with a Taylor's series expansion of $\displaystyle ln(1+ \mu)$.
$\log_e(1+\mu) = \mu - \dfrac{\mu^2}{2} + \dfrac{\mu^3}{3} - \dfrac{\mu^4}{4}+\cdots$ and plug that in,
to get $\log_e(1+\mu) =\dfrac12- \dfrac\mu3+\dfrac{\mu^2}{4}+\cdots.$
now, comparing $\log_e(1+\mu)$ with $t_h(\mu)=1-\frac{\mu}{2}+\cdots$
what can I say?

4. Re: Estimate oredr O(h)

Originally Posted by grandy
Consider non-dimensional differential equation for the height at the highest point is given by
$$h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} log_e(1+\mu)$$
$0<\mu<<1.$
Deduce an estimate to $O(\mu)$ for $h(\mu)$ and compare with $t_h(\mu)=1-\frac{\mu}{2}+...$
well done................
Soran University