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Telescoping Sum — Definition, Formula & Examples

A telescoping sum is a series whose partial sums simplify because most terms cancel with adjacent terms, leaving only the first and last few terms. The name comes from the way the sum collapses inward, like a collapsing telescope.

A series ∑n=1N(an−an+1)\sum_{n=1}^{N}(a_n - a_{n+1}) is called telescoping because its partial sum reduces to a1−aN+1a_1 - a_{N+1}. If lim⁡N→∞aN+1\lim_{N\to\infty} a_{N+1} exists, the infinite series converges to a1−lim⁡N→∞aN+1a_1 - \lim_{N\to\infty} a_{N+1}.

Key Formula

∑n=1N(an−an+1)=a1−aN+1\sum_{n=1}^{N}(a_n - a_{n+1}) = a_1 - a_{N+1}
Where:
  • ana_n = The sequence whose consecutive differences form the series terms
  • NN = Upper index of the partial sum

How It Works

To evaluate a telescoping series, first decompose the general term into a difference of two simpler expressions — typically using partial fraction decomposition. Then write out several consecutive terms of the partial sum and observe the cancellation pattern. Most intermediate terms appear once with a positive sign and once with a negative sign, so they vanish. The partial sum collapses to just a few surviving terms, and you take the limit as N→∞N \to \infty to find the value of the series.

Worked Example

Problem: Evaluate the infinite series ∑n=1∞1n(n+1)\displaystyle\sum_{n=1}^{\infty} \frac{1}{n(n+1)}.
Partial Fractions: Decompose the general term into a telescoping difference.
1n(n+1)=1n−1n+1\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}
Write the Partial Sum: Expand the first several terms and observe the cancellation.
SN=(11−12)+(12−13)+⋯+(1N−1N+1)=1−1N+1S_N = \left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\cdots+\left(\frac{1}{N}-\frac{1}{N+1}\right) = 1 - \frac{1}{N+1}
Take the Limit: Let N→∞N \to \infty to find the series value.
lim⁡N→∞SN=1−0=1\lim_{N\to\infty} S_N = 1 - 0 = 1
Answer: The series converges to 11.

Why It Matters

Telescoping sums appear frequently in Calculus II when you need the exact value of a series, not just whether it converges. They also arise in discrete mathematics and algorithm analysis, where summing differences of consecutive terms is a standard technique analogous to the Fundamental Theorem of Calculus.

Common Mistakes

Mistake: Assuming every series with partial fractions will telescope.
Correction: A partial fraction decomposition only produces a telescoping sum when the resulting terms cancel consecutively. Always write out several terms explicitly to verify the cancellation pattern before simplifying.