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Partial Sums — Definition, Formula & Examples

A partial sum is the result of adding up the first nn terms of a series. By examining what happens to partial sums as nn grows, you can determine whether an infinite series converges to a finite value or diverges.

Given an infinite series ∑k=1∞ak\displaystyle\sum_{k=1}^{\infty} a_k, the nn-th partial sum is defined as Sn=∑k=1nak=a1+a2+⋯+anS_n = \displaystyle\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n. The series converges to a sum SS if and only if the sequence of partial sums {Sn}\{S_n\} has a finite limit: lim⁡n→∞Sn=S\displaystyle\lim_{n \to \infty} S_n = S. If no such finite limit exists, 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 of the series
  • aka_k = The $k$-th term of the series
  • nn = The number of terms included in the sum

How It Works

To use partial sums, you compute S1=a1S_1 = a_1, then S2=a1+a2S_2 = a_1 + a_2, then S3=a1+a2+a3S_3 = a_1 + a_2 + a_3, and so on. Each partial sum builds on the previous one: Sn=Sn−1+anS_n = S_{n-1} + a_n. You then analyze the sequence S1,S2,S3,…S_1, S_2, S_3, \ldots to see if it approaches a specific number. If the partial sums settle toward a limit, the series converges to that value; if they grow without bound or oscillate indefinitely, the series diverges. This approach is the foundational definition of what it means for an infinite series to have a sum.

Worked Example

Problem: Find the first four partial sums of the geometric series ∑k=1∞(12)k\displaystyle\sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^k and determine what value the series converges to.
Step 1: Compute the first partial sum by evaluating the first term.
S1=12S_1 = \frac{1}{2}
Step 2: Add the second term to get the second partial sum.
S2=12+14=34S_2 = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}
Step 3: Continue accumulating terms for the third and fourth partial sums.
S3=34+18=78,S4=78+116=1516S_3 = \frac{3}{4} + \frac{1}{8} = \frac{7}{8}, \quad S_4 = \frac{7}{8} + \frac{1}{16} = \frac{15}{16}
Step 4: Observe the pattern: Sn=1−(12)nS_n = 1 - \left(\frac{1}{2}\right)^n. As n→∞n \to \infty, the term (12)n→0\left(\frac{1}{2}\right)^n \to 0, so the partial sums approach 1.
lim⁡n→∞Sn=lim⁡n→∞(1−12n)=1\lim_{n \to \infty} S_n = \lim_{n \to \infty} \left(1 - \frac{1}{2^n}\right) = 1
Answer: The partial sums are 12,34,78,1516,…\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \ldots and the series converges to 11.

Another Example

Problem: Determine whether the series ∑k=1∞(−1)k+1=1−1+1−1+⋯\displaystyle\sum_{k=1}^{\infty} (-1)^{k+1} = 1 - 1 + 1 - 1 + \cdots converges by examining its partial sums.
Step 1: Write out the first several partial sums.
S1=1,S2=0,S3=1,S4=0S_1 = 1, \quad S_2 = 0, \quad S_3 = 1, \quad S_4 = 0
Step 2: The partial sums alternate between 1 and 0 forever. Since {Sn}\{S_n\} does not approach a single value, the limit does not exist.
lim⁡n→∞Sn does not exist\lim_{n \to \infty} S_n \text{ does not exist}
Answer: The series diverges because the sequence of partial sums oscillates and has no limit.

Visualization

Why It Matters

Partial sums are central to AP Calculus BC, where nearly every convergence test relies on the behavior of the sequence {Sn}\{S_n\}. Engineers and physicists use partial sums to approximate values of functions represented by power series, stopping at enough terms to reach a desired accuracy. Understanding partial sums also prepares you for topics like Fourier series in advanced mathematics and signal processing.

Common Mistakes

Mistake: Confusing the sequence of terms {an}\{a_n\} with the sequence of partial sums {Sn}\{S_n\}.
Correction: Remember that ana_n is a single term, while SnS_n is the cumulative total of the first nn terms. The fact that an→0a_n \to 0 does not guarantee that SnS_n converges (consider the harmonic series).
Mistake: Assuming a series converges just because the partial sums grow slowly.
Correction: Slow growth is still divergence. For example, the harmonic series partial sums increase without bound even though they do so very gradually. You must verify that lim⁡n→∞Sn\lim_{n \to \infty} S_n is a finite number.

Related Terms