nth Partial Sum

nth Partial Sum

The sum of the first n terms of an infinite series.

Example showing partial sums S1=1, S2=1+1/4, S3=1+1/4+1/9, S4=1+1/4+1/9+1/16 for series 1+1/4+1/9+1/16+…+1/n²

Key Formula

S_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n

Where:

$S_n$ = The nth partial sum — the total when you add the first n terms
$a_k$ = The kth term of the series
$n$ = The number of terms being added
$k$ = The index of summation, running from 1 to n

Worked Example

Problem: Find the 5th partial sum of the series whose terms are given by a_k = 2k.

Step 1: Identify the first 5 terms by substituting k = 1, 2, 3, 4, 5 into a_k = 2k.

a_1 = 2,\quad a_2 = 4,\quad a_3 = 6,\quad a_4 = 8,\quad a_5 = 10

Step 2: Write out the partial sum formula for n = 5.

S_5 = a_1 + a_2 + a_3 + a_4 + a_5

Step 3: Substitute the values and add them up.

S_5 = 2 + 4 + 6 + 8 + 10 = 30

Answer: The 5th partial sum is S₅ = 30.

Another Example

This example uses a geometric series, showing how a closed-form partial sum formula can replace term-by-term addition.

Problem: Find the 4th partial sum of the geometric series where a_k = 3 · (1/2)^(k−1).

Step 1: List the first 4 terms by substituting k = 1, 2, 3, 4.

a_1 = 3,\quad a_2 = \frac{3}{2},\quad a_3 = \frac{3}{4},\quad a_4 = \frac{3}{8}

Step 2: Use the geometric partial sum formula with first term a = 3, common ratio r = 1/2, and n = 4.

S_n = a\,\frac{1 - r^n}{1 - r} = 3\,\frac{1 - (1/2)^4}{1 - 1/2}

Step 3: Compute the numerator: 1 − (1/2)⁴ = 1 − 1/16 = 15/16.

S_4 = 3 \cdot \frac{15/16}{1/2} = 3 \cdot \frac{15}{8}

Step 4: Multiply to get the final answer.

S_4 = \frac{45}{8} = 5.625

Answer: The 4th partial sum is S₄ = 45/8 = 5.625.

Frequently Asked Questions

What is the difference between a partial sum and an infinite series?

A partial sum adds only a finite number of terms (the first n), giving you a definite numerical value. An infinite series represents the sum of all infinitely many terms, which may or may not have a finite value. The infinite series equals the limit of the partial sums as n approaches infinity, provided that limit exists.

How do you use partial sums to determine if a series converges?

You form the sequence of partial sums S₁, S₂, S₃, … and examine whether this sequence approaches a finite limit L as n → ∞. If lim(n→∞) Sₙ = L exists and is finite, the series converges to L. If the limit does not exist or is infinite, the series diverges.

Is there a shortcut formula for the nth partial sum?

Shortcut formulas exist for specific types of series. For an arithmetic series, Sₙ = n(a₁ + aₙ)/2. For a geometric series, Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1. For a general series, there may be no closed-form shortcut, and you must add the terms directly or use telescoping techniques.

nth Partial Sum vs. Sum of an Infinite Series

	nth Partial Sum	Sum of an Infinite Series
Definition	Sum of the first n terms: S_n = a₁ + a₂ + … + aₙ	Sum of all terms: S = a₁ + a₂ + a₃ + …
Number of terms	Finite (exactly n terms)	Infinitely many terms
Always has a value?	Yes — adding finitely many numbers always gives a result	No — only if the series converges
Relationship	Each S_n is one entry in the sequence of partial sums	Equals lim(n→∞) S_n, if this limit exists

Why It Matters

Partial sums are the primary tool for defining convergence in calculus and analysis — you cannot rigorously discuss whether an infinite series has a finite value without them. They appear in Taylor series approximations, where the nth partial sum gives a polynomial approximation to functions like eˣ, sin x, and ln(1 + x). In applied settings, partial sums let you compute practical approximations when summing all terms is impossible.

Common Mistakes

Mistake: Confusing the nth term aₙ with the nth partial sum Sₙ.

Correction: The nth term aₙ is a single value in the sequence, while the nth partial sum Sₙ is the total of all terms from a₁ through aₙ. For example, if aₙ = 2n, then a₃ = 6 but S₃ = 2 + 4 + 6 = 12.

Mistake: Starting the index at the wrong value (e.g., summing from k = 0 when the series starts at k = 1, or vice versa).

Correction: Always check the starting index of the series. A series that begins at k = 0 includes the term a₀, so S_n would be a₀ + a₁ + … + aₙ, which is actually n + 1 terms. Match your partial sum to the series definition exactly.

Related Terms

Sum — General concept of adding numbers together
Term — Each individual element aₖ in the series
Infinite Series — The full sum whose partial sums approximate it
Convergence Tests — Methods that use partial sums to test convergence
Sequence of Partial Sums — The ordered list S₁, S₂, S₃, … of all partial sums
Geometric Series — A series type with a closed-form partial sum formula
Arithmetic Series — A series type with a simple partial sum formula
Telescoping Series — A series whose partial sums simplify by cancellation