Geometric and Telescoping Series

A geometric series is simply a sum where each term is the same fraction of the one before, like a ball bouncing back to 60% of its last height every time. The name comes from the geometric mean: each term is the geometric mean of its two neighbors, since $ar^n = \sqrt{ar^{n-1}\cdot ar^{n+1}}$ (for positive terms), the way each term of an arithmetic sequence is the ordinary average of its neighbors. They're good because they're one of the very few kinds of series you can add up exactly, and they show up everywhere from repeating decimals to drug doses. A telescoping series, the other kind here, collapses like a folding telescope because nearly every term cancels. The formula $\sum_{n=0}^{\infty} a r^n = \frac{a}{1-r}$ looks like a mouthful, but it's just two numbers: $a$ (the first term, where you start) and $r$ (the fraction you multiply by each time). The condition $|r| < 1$ just means the terms really are shrinking. That's the whole formula: first term over one minus the ratio.

You've already met a few of these. The bouncing ball from the series lesson and $\sum \frac{1}{2^n}$ are both geometric. Most series can't be summed exactly at all, so it's worth seeing why this family can.

Why the formula holds. Write out $s_n = a + ar + \cdots + ar^n$ and multiply it by $r$. You get $rs_n = ar + ar^2 + \cdots + ar^{n+1}$, which is the same list shifted over by one. Subtract, and almost everything cancels, leaving $s_n - rs_n = a - ar^{n+1}$. If $|r| < 1$, then $r^{n+1} \to 0$ and $s_n \to \frac{a}{1-r}$. If $|r| \ge 1$, the terms don't go to 0, so the series diverges by the divergence test.

How to use it. The sum is $\dfrac{\text{first term}}{1 - \text{ratio}}$, whatever the starting index.

A common mistake. It's tempting to plug in $a$ from the formula $ar^n$ even when the series starts at $n = 1$. That $a$ is only the first term when you start at $n = 0$. In $\sum_{n=1}^{\infty} 3\left(\frac25\right)^n$ the first term is $3\cdot\frac25$, not $3$. Always write down the actual first term you're adding, and the ratio, and then you're done.

Repeating decimals are geometric series:

Telescoping series. Here's the other kind you can sum exactly. If each term is a difference $b_n - b_{n+1}$, each term's second half cancels the next term's first half, and the partial sums collapse like a folding telescope: $s_n = b_1 - b_{n+1}$. Partial fractions, the same trick you used for integrals, are usually how you spot this. For example, $\frac{1}{n(n+1)} = \frac1n - \frac{1}{n+1}$, so

The harmonic series $\sum \frac1n$ diverges, as the last lesson showed. Hang on to all three: geometric, telescoping and harmonic series are the yardsticks you'll measure other series against for the rest of this unit.

Geometric series price annuities and perpetuities, give the total distance of a bouncing ball, and predict a drug's steady-state level under repeated doses: each dose leaves a fixed fraction behind for the next.