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Partial Sum — Definition, Formula & Examples

A partial sum is the result of adding up the first nn terms of a sequence. As you increase nn, the sequence of partial sums tells you whether an infinite series converges or diverges.

Given a sequence {ak}\{a_k\}, the nn-th partial sum is defined as Sn=∑k=1nakS_n = \sum_{k=1}^{n} a_k. The infinite series ∑k=1∞ak\sum_{k=1}^{\infty} a_k converges to LL if and only if lim⁡n→∞Sn=L\lim_{n \to \infty} S_n = L; otherwise the series diverges.

Key Formula

Sn=∑k=1nak=a1+a2+a3+⋯+anS_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n
Where:
  • SnS_n = The $n$-th partial sum
  • aka_k = The $k$-th term of the sequence
  • nn = The number of terms being summed

How It Works

To find a partial sum, simply add the first nn terms of your sequence. Each value of nn gives a different partial sum, forming a new sequence S1,S2,S3,…S_1, S_2, S_3, \ldots called the sequence of partial sums. If this sequence approaches a finite limit as n→∞n \to \infty, the series converges to that limit. If the partial sums grow without bound or oscillate indefinitely, the series diverges. Partial sums are the bridge between finite addition and infinite series.

Worked Example

Problem: Find the first four partial sums of the series ∑k=1∞12k\sum_{k=1}^{\infty} \frac{1}{2^k} and identify what value the series appears to approach.
Step 1: Compute S1S_1 by taking just the first term.
S1=12=0.5S_1 = \frac{1}{2} = 0.5
Step 2: Add the second term to get S2S_2.
S2=12+14=34=0.75S_2 = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} = 0.75
Step 3: Continue for S3S_3 and S4S_4.
S3=12+14+18=78=0.875S4=12+14+18+116=1516=0.9375S_3 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{7}{8} = 0.875 \qquad S_4 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = \frac{15}{16} = 0.9375
Answer: The partial sums 0.5,0.75,0.875,0.9375,…0.5, 0.75, 0.875, 0.9375, \ldots are approaching 11. Indeed, this geometric series converges to 1/21−1/2=1\frac{1/2}{1 - 1/2} = 1.

Visualization

Why It Matters

Partial sums are the fundamental tool for determining whether an infinite series converges. In AP Calculus BC and college-level calculus, nearly every convergence test ultimately relies on the behavior of partial sums. They also appear in numerical methods and physics when you approximate an infinite process by summing finitely many terms.

Common Mistakes

Mistake: Confusing the nn-th term ana_n with the nn-th partial sum SnS_n.
Correction: ana_n is a single term of the sequence, while SnS_n is the cumulative total of the first nn terms. The fact that an→0a_n \to 0 does not guarantee SnS_n converges (e.g., the harmonic series).