Telescoping Series

What is a telescoping series?

A telescoping series is one where most terms cancel in pairs when you write them out, leaving only the first and last. The name comes from a collapsible telescope — the middle sections disappear.

Step-by-step: Evaluate $\sum_{k=1}^{n} \left[\frac{1}{k} - \frac{1}{k+1}\right]$

Step 1 — Write out the first several terms:

$\left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right)$

Step 2 — Notice the cancellations: The $-1/2$ from the first term cancels with the $+1/2$ from the second. The $-1/3$ from the second cancels with the $+1/3$ from the third. And so on.

Step 3 — After all cancellations, only two pieces survive: the very first positive term $\frac{1}{1}$ and the very last negative term $-\frac{1}{n+1}$.

$S = 1 - \frac{1}{n+1} = \frac{n}{n+1}$

Step 4 — Check with $n = 3$: $(1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) = 1 - 1/4 = 3/4$. Formula: $3/4$ ✓.

As $n \to \infty$

$\lim_{n \to \infty} \frac{n}{n+1} = 1$

The infinite series converges to 1.

How to recognize telescoping

If each term looks like $f(k) - f(k+1)$ (a difference of consecutive values of some function), the series telescopes. Common forms include partial fractions like $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$.

Try it in the visualization

Watch terms cancel pair by pair as $n$ increases. The partial sums converge toward 1. Color-coded terms show which positive and negative parts cancel.