Calculus II · Unit 4A · lesson

Telescoping Series

Concept

Learning objectives

use partial fractions or algebraic decomposition to expose cancellation and compute a series sum.

Telescoping Series

Explanation

Write the partial sum before trusting the cancellation

A telescoping series contains terms arranged so that most contributions cancel in a finite partial sum. The cancellation is not visible from the infinite notation alone; it appears after several terms are written with their signs. This is why a correct solution displays $s_N$ rather than announcing that "everything cancels."

After cancellation, a few boundary terms remain. The series converges if those remaining terms approach a finite limit. Partial fractions frequently create telescoping structure, especially for rational terms involving consecutive factors such as $n(n+1)$ or $(n+a)(n+b)$ .

Bridge

Cancellation is a fact about partial sums

A telescoping series is not a mysterious new species of infinite sum. It is an ordinary series whose finite partial sums simplify because most terms cancel. The safe procedure is therefore to write several terms of $s_N$ , mark the cancellation, and keep the boundary terms that survive. Only after the finite formula is correct do we let $N\to\infty$ .

Partial fractions often reveal the hidden cancellation. The important question is not merely whether adjacent-looking symbols resemble one another, but whether the shifted indices line up across the entire finite sum. A gap of two or three indices may leave several terms at each boundary rather than just one.

**Interior terms cancel; boundary terms survive**
A finite telescoping partial sum is displayed as shifted positive and negative rows. Matching interior terms cancel, leaving only terms at the two ends.
Interior terms cancel; boundary terms survive. A cancellation chain showing interior terms disappearing while boundary terms remain.

Proof idea

The limit enters only after the algebra is finite

For every fixed $N$ , cancellation is ordinary finite algebra. Once $s_N$ has been reduced to its surviving terms, convergence is determined by the limit of that finite formula.

Concept

Telescoping workflow

Decompose the term, write a finite partial sum, cancel only terms actually present, simplify the boundary expression, and then let $N\to\infty$ .

Guided walkthrough

A standard telescoping sum

Since

\frac1{n(n+1)}=\frac1n-\frac1{n+1},

we have

s_N=\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\cdots+\left(\frac1N-\frac1{N+1}\right)=1-\frac1{N+1}.

Thus the infinite series sums to $1$ .

Worked example

Telescoping with a two-step shift

Evaluate

\sum_{n=1}^{\infty}\frac{1}{n(n+2)}.

First decompose

\frac{1}{n(n+2)}=\frac12\left(\frac1n-\frac1{n+2}\right).

Then

s_N=\frac12\left(1+\frac12-\frac1{N+1}-\frac1{N+2}\right).

The final two fractions vanish as $N\to\infty$ , so

\sum_{n=1}^{\infty}\frac{1}{n(n+2)}=\frac12\left(1+\frac12\right)=\frac34.

Notice that two initial terms survive because the shift is two.

Common mistake

Never cancel inside an unwritten infinite expression

Cancellation must be demonstrated in the finite partial sum. Informally crossing out infinitely many terms can hide boundary terms or invalid rearrangements.

Interactive checku4a-telescoping_series-01

Evaluate $\sum_{n=1}^{\infty}\frac1{n(n+1)}$ .

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Use $1/[n(n+1)]=1/n-1/(n+1)$ .

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Exercise

Evaluate $\sum_{n=1}^{\infty}2/[(n+1)(n+3)]$ .

Exercise

Write the first five terms before canceling.

Exercise

Give a telescoping series that diverges because a boundary term fails to settle.

Exercise

Explain why cancellation in an infinite expression must be justified through finite partial sums.

After the explanation

Use the section idea

Reading lens

Build every infinite sum from finite partial sums, and expose geometric or telescoping structure before taking a limit.

Mental model

A series converges exactly when its sequence of partial sums approaches a finite value.

Decision

Check the term limit first, then look for an exact partial-sum pattern before selecting a comparison test.

Common trap

Concluding convergence from terms approaching zero or canceling inside an unwritten infinite expression.

Check yourself

Can you write the relevant finite partial sum and identify which terms survive?

Source & rights

Original instruction with traceable references.

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