Algebra · step 7 of 13

Summation notation Σ & telescoping

Σ notation compresses a sum into an index, a range and a term. The manipulations you need are few but must be exact: constants factor out, sums split over addition, and shifting the index requires compensating in the term. Telescoping is the star technique — if the term can be written as f(k) − f(k+1), almost everything cancels and the sum collapses to two values.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Expand the first and last termsWriting out k = 1, k = 2 and k = n makes the structure visible and catches off-by-one errors.
Factor constants outΣ c·aₖ = c·Σ aₖ, and Σ (aₖ + bₖ) = Σ aₖ + Σ bₖ. Note Σ aₖbₖ does NOT split.
Shift indices carefullyReplacing k by k+1 changes both the limits and the term. Substitute and re-derive rather than guessing.
Telescope by partial fractionsSplit the term into a difference of consecutive values; all the middle terms cancel in pairs.

Worked example

Evaluate Σ_{k=1}^{n} 1/(k(k+1)).

Partial fractions: 1/(k(k+1)) = 1/k − 1/(k+1).
The sum becomes (1/1 − 1/2) + (1/2 − 1/3) + … + (1/n − 1/(n+1)).
Every interior term appears once positive and once negative and cancels.
Only the first and last survive: 1 − 1/(n+1).

Answer. n/(n+1) — and letting n → ∞ shows the infinite series converges to 1.

Where marks get dropped

These are the specific errors that cost credit on summation notation σ & telescoping questions — QED's rubric penalises each of them separately.

Splitting Σ aₖbₖ into (Σaₖ)(Σbₖ). Sums do not distribute over products.
Off-by-one errors when shifting the index. Always rewrite the first and last terms explicitly after a shift.
Cancelling telescoping terms without checking the ends. The leftover is exactly the terms not paired up.

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Summation notation Σ & telescoping — frequently asked questions

What are the standard sum formulas?

Σk = n(n+1)/2, Σk² = n(n+1)(2n+1)/6, Σk³ = [n(n+1)/2]². The last is the square of the first, which is a memorable coincidence.

How do I spot a telescoping sum?

Look for a term that is a rational function with factors differing by a constant, or a difference of consecutive function values. Partial fractions will reveal it.

Does the index letter matter?

No — it is bound, so Σ_k aₖ and Σ_j a_j are identical. But do not reuse a letter that is already in use outside the sum.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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