Partial Sums

Single-variable calculus from first principles

Take a sequence and start adding the terms up as you go. After one term you have a₁. After two, a₁ + a₂. After three, a₁ + a₂ + a₃. Each of these running totals is called a partial sum, written Sₙ — the sum of the first n terms.

The partial sums themselves form a new sequence (S₁, S₂, S₃, …), and we can ask the very same question as last lesson: does this running total settle onto a limit? If it does, we call that limit the sum of the series.

Picture a tip jar you keep topping up: each running total is a partial sum, the money in the jar after the latest contribution. If every contribution is half the size of the one before — like adding 1/2 + 1/4 + 1/8 + … of a dollar — the jar fills quickly at first, then barely rises, hugging a ceiling. That ceiling it never quite passes is the sum of the series, here exactly 1 dollar.

Where this lives in MLPartial sums are everywhere in machine learning. Cumulative training loss is a running total over steps. In reinforcement learning, a discounted return is literally a geometric series — future rewards multiplied by a ratio γ < 1 each step — and the formula 1/(1 − γ) tells you the largest total reward possible.

▶ Partial Sums

← Sequences Bridge to Integration →