Mathwords logoReference LibraryMathwords

Partial Sum of a Series

Partial Sum of a Series

The sum of a finite number of terms of a series.

 

 

See also

nth partial sum, convergence tests, sequence of partial sums

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 nth partial sum — the sum of the first n terms
  • aka_k = The kth term of the series
  • nn = The number of terms being added

Worked Example

Problem: Find the 4th partial sum of the series 2 + 5 + 8 + 11 + 14 + 17 + …
Step 1: Identify the first 4 terms of the series.
a1=2,a2=5,a3=8,a4=11a_1 = 2, \quad a_2 = 5, \quad a_3 = 8, \quad a_4 = 11
Step 2: Add those 4 terms together to form the 4th partial sum.
S4=2+5+8+11S_4 = 2 + 5 + 8 + 11
Step 3: Compute the result.
S4=26S_4 = 26
Answer: The 4th partial sum is S₄ = 26.

Another Example

Problem: Find the 5th partial sum of the geometric series where each term is (1/2)^k, starting at k = 0.
Step 1: Write out the first 5 terms (k = 0 through k = 4).
a0=1,a1=12,a2=14,a3=18,a4=116a_0 = 1, \quad a_1 = \tfrac{1}{2}, \quad a_2 = \tfrac{1}{4}, \quad a_3 = \tfrac{1}{8}, \quad a_4 = \tfrac{1}{16}
Step 2: Add the terms together.
S5=1+12+14+18+116S_5 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}
Step 3: Use a common denominator of 16 to combine.
S5=16+8+4+2+116=3116S_5 = \frac{16 + 8 + 4 + 2 + 1}{16} = \frac{31}{16}
Answer: The 5th partial sum is S₅ = 31/16 = 1.9375.

Frequently Asked Questions

What is the difference between a partial sum and the sum of a series?
A partial sum adds only a finite number of terms (the first n terms), giving you a specific number for each n. The sum of a series (if it exists) is the limit that the partial sums approach as n goes to infinity. If that limit exists and is finite, the series converges to that sum.
How do partial sums tell you whether a series converges or diverges?
You form the sequence of partial sums S₁, S₂, S₃, … and examine its behavior as n grows. If this sequence approaches a finite limit L, the series converges to L. If the sequence grows without bound or oscillates without settling down, the series diverges.

Partial Sum vs. Infinite Series Sum

A partial sum SnS_n is a single finite number obtained by adding the first nn terms. The sum of an infinite series is the limit lim⁡n→∞Sn\lim_{n \to \infty} S_n, which may or may not exist. Every infinite series has partial sums, but only convergent series have a finite total sum.

Why It Matters

Partial sums are the main tool for determining whether an infinite series converges. By studying how the sequence S1,S2,S3,…S_1, S_2, S_3, \ldots behaves, you can decide if the series has a finite sum or not. They also appear in practical applications—whenever you approximate an infinite process (like computing π\pi or ee from a series), you are actually calculating a partial sum and using it as an approximation.

Common Mistakes

Mistake: Confusing the nth term ana_n with the nth partial sum SnS_n.
Correction: ana_n is a single term of the series, while Sn=a1+a2+⋯+anS_n = a_1 + a_2 + \cdots + a_n is the cumulative total of the first n terms. For example, if a4=11a_4 = 11, that does not mean S4=11S_4 = 11.
Mistake: Thinking that if an→0a_n \to 0, then the partial sums must converge.
Correction: Having terms shrink to zero is necessary but not sufficient for convergence. The harmonic series ∑1/n\sum 1/n has terms going to zero, yet its partial sums grow without bound (diverge).

Related Terms

  • Series — The infinite sum whose partial sums are computed
  • nth Partial Sum — The specific partial sum using the first n terms
  • Sequence of Partial Sums — The sequence S₁, S₂, S₃, … formed from partial sums
  • Convergence Tests — Methods that use partial sum behavior to test convergence
  • Sum — General operation of adding quantities together
  • Finite — Partial sums always involve finitely many terms
  • Term — Each individual element being added in the series