A student asks:

When I do inequalities with geometric series, I sometimes get my inequality sign the wrong way round - I can usually fudge it, but I’d like to be getting it right.

OK, I confess, that student is me, circa 1995. And, in fact, I only figured it out once I started teaching.

If I was asked something like this, from a 2008 paper ((obviously, I wouldn’t have been in 1995, but you know what I mean)):

A geometric series has first term 5 and common ratio $\frac45$. Given that the sum to $k$ terms of the series is greater than 24.95, find the smallest possible value of $k$.

… I would have had a decent stab at it - but I’d have dropped some marks.

Here’s how I’d do it now

These days, I know the formula for the sum of a geometric series, but I’d have looked it up back then; it’s $S_n = \frac{a(1-r^n)}{1-r}$. Here, we know $a$ and $r$, so we can throw them in:

$S_k = \frac{5\left(1 - \left(\frac45\right)^n\right)}{1-\left( \frac45 \right) } > 24.95$

Wow, what a mess. The bottom tidies up nicely to give $\frac{1}{5}$, and dividing by $\frac15$ is the same as multiplying by $5$. That gives:

$25\left(1 - \left(\frac45\right)^k\right) > 24.95$, or $1 - \left(\frac45\right)^k > 0.998$, or better still $0.002 > \left(\frac45\right)^k$

Naturally, I’d take logs here:

$\log(0.002) > k\log(0.8)$

And here’s where I’d go wrong. I’d divide by $\log(0.8)$, but I’d forget the critical thing:

When you divide by a negative number, the inequality changes direction

So, dividing by $\log(0.8)$ gives:

$\frac{\log( 0.002)}{\log(0.8)} < k$ - notice that the inequality is pointing the other way. I know $\log(0.8)$ is negative, because $\log(1)=0$ in any base - and any smaller argument gives a negative number.

… and then I’d have worked it out on a calculator. These days? I’d say it was $\ln(500)$ divided by $\ln(5) - \ln(4)$ ((why?)), which is about $(1.6 + 2.3 + 2.3) = 6.2$ divided by about $1.6 - 1.4 = 0.2$, so somewhere in the region of 34. (It’s actually 27.9 - I underestimated the bottom by about 10%. Don’t tell the Mathematical Ninja!)