Mathwords: Sequence of Partial Sums

Q: What is the difference between a series and a sequence of partial sums?

A series is the expression $\sum_{k=1}^{\infty} a_k$, which represents the idea of adding infinitely many terms. The sequence of partial sums $\{S_1, S_2, S_3, \ldots\}$ is the actual sequence you build by computing finite sums one term at a time. The series converges if and only if this sequence of partial sums converges to a finite limit.

Q: How do you tell if a series converges using partial sums?

Compute (or find a formula for) the nth partial sum $S_n$, then evaluate $\lim_{n \to \infty} S_n$. If this limit exists and is a finite number $L$, the series converges to $L$. If the limit does not exist or is infinite, the series diverges.

Q: Can you always find a formula for the nth partial sum?

No. Closed-form expressions for $S_n$ exist for certain types of series such as geometric series, telescoping series, and a few others. For most series, you cannot write $S_n$ in a simple closed form, which is why convergence tests (like the ratio test or comparison test) are so important — they let you determine convergence without needing an explicit formula for $S_n$.

Key Formula

\{S_n\} = \{S_1,\, S_2,\, S_3,\, \ldots\} \quad \text{where} \quad 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 sum of the first n terms of the series
$a_k$ = The kth term of the original series
$n$ = The number of terms being summed (a positive integer)

Worked Example

Problem: Find the first five terms of the sequence of partial sums for the series

\sum_{k=1}^{\infty} \frac{1}{2^k}

.

Step 1: Identify the terms of the series. The kth term is

a_k = \frac{1}{2^k}

, so the first several terms are

\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \ldots

a_1 = \tfrac{1}{2},\quad a_2 = \tfrac{1}{4},\quad a_3 = \tfrac{1}{8},\quad a_4 = \tfrac{1}{16},\quad a_5 = \tfrac{1}{32}

Step 2: Compute

S_1

, the first partial sum.

S_1 = a_1 = \frac{1}{2}

Step 3: Compute

S_2

and

S_3

by adding the next term each time.

S_2 = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}, \qquad S_3 = \frac{3}{4} + \frac{1}{8} = \frac{7}{8}

Step 4: Continue for

S_4

and

S_5

.

S_4 = \frac{7}{8} + \frac{1}{16} = \frac{15}{16}, \qquad S_5 = \frac{15}{16} + \frac{1}{32} = \frac{31}{32}

Step 5: Write the sequence of partial sums. Notice each term follows the pattern

S_n = 1 - \frac{1}{2^n}

, and as

n \to \infty

,

S_n \to 1

. So the series converges to 1.

\{S_n\} = \left\{\frac{1}{2},\; \frac{3}{4},\; \frac{7}{8},\; \frac{15}{16},\; \frac{31}{32},\; \ldots \right\}

Answer: The first five partial sums are

\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \frac{31}{32}

. The sequence converges to 1.

Another Example

This example shows a telescoping series where you can find a closed-form formula for $S_n$ and then take its limit, rather than just listing numerical values of partial sums.

Problem: Find a closed-form expression for the nth partial sum of the telescoping series

\sum_{k=1}^{\infty} \left(\frac{1}{k} - \frac{1}{k+1}\right)

and determine whether the series converges.

Step 1: Write out the partial sum

S_n

by expanding the first several terms.

S_n = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right)

Step 2: Most terms cancel in pairs (this is why it is called a telescoping series). After cancellation, only the first and last parts remain.

S_n = 1 - \frac{1}{n+1}

Step 3: Check a few values:

S_1 = 1 - \frac{1}{2} = \frac{1}{2}

,

S_2 = 1 - \frac{1}{3} = \frac{2}{3}

,

S_3 = 1 - \frac{1}{4} = \frac{3}{4}

. These match direct computation.

\{S_n\} = \left\{\frac{1}{2},\; \frac{2}{3},\; \frac{3}{4},\; \frac{4}{5},\; \ldots\right\}

Step 4: Take the limit of the sequence of partial sums to determine convergence.

\lim_{n \to \infty} S_n = \lim_{n \to \infty} \left(1 - \frac{1}{n+1}\right) = 1

Answer: The nth partial sum is

S_n = 1 - \frac{1}{n+1}

, and the series converges to 1.

Frequently Asked Questions

What is the difference between a series and a sequence of partial sums?

A series is the expression

\sum_{k=1}^{\infty} a_k

, which represents the idea of adding infinitely many terms. The sequence of partial sums

\{S_1, S_2, S_3, \ldots\}

is the actual sequence you build by computing finite sums one term at a time. The series converges if and only if this sequence of partial sums converges to a finite limit.

How do you tell if a series converges using partial sums?

Compute (or find a formula for) the nth partial sum

S_n

, then evaluate

\lim_{n \to \infty} S_n

. If this limit exists and is a finite number

L

, the series converges to

L

. If the limit does not exist or is infinite, the series diverges.

Can you always find a formula for the nth partial sum?

No. Closed-form expressions for

S_n

exist for certain types of series such as geometric series, telescoping series, and a few others. For most series, you cannot write

S_n

in a simple closed form, which is why convergence tests (like the ratio test or comparison test) are so important — they let you determine convergence without needing an explicit formula for

S_n

.

Sequence of Partial Sums vs. nth Partial Sum

	Sequence of Partial Sums	nth Partial Sum
What it is	The entire sequence $\{S_1, S_2, S_3, \ldots\}$	A single value $S_n$ for one particular $n$
Formula	$\{S_n\}_{n=1}^{\infty}$ where each $S_n = \sum_{k=1}^{n} a_k$	$S_n = a_1 + a_2 + \cdots + a_n$
Purpose	Analyze convergence by studying the behavior of the whole sequence	Approximate the sum of a series using the first $n$ terms
Example output	$\{\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \ldots\}$ (a list of values)	$S_3 = \frac{7}{8}$ (a single number)

Why It Matters

The sequence of partial sums is the fundamental tool for defining what it means for an infinite series to converge. In calculus courses, you encounter it whenever you study geometric series, telescoping series, or power series representations of functions. Understanding this concept bridges the gap between finite addition (which you already know) and the idea of summing infinitely many terms.

Common Mistakes

Mistake: Confusing the original sequence

\{a_n\}

with the sequence of partial sums

\{S_n\}

.

Correction: The terms

a_n

are the individual pieces being added. The partial sums

S_n

are cumulative totals. For example, if

a_n = \frac{1}{2^n}

, then

a_3 = \frac{1}{8}

but

S_3 = \frac{7}{8}

. These are very different sequences.

Mistake: Assuming that if

a_n \to 0

then

S_n

must converge to a finite limit.

Correction: The terms approaching zero is necessary but not sufficient for convergence. The harmonic series

\sum \frac{1}{k}

has

a_k \to 0

, yet

S_n \to \infty

. You must examine the limit of

S_n

itself (or use a convergence test) to determine whether the series converges.

Related Terms

Sequence — A sequence of partial sums is itself a sequence
nth Partial Sum — Each individual term in the sequence of partial sums
Series — The sum whose convergence is determined by partial sums
Convergence Tests — Methods used when closed-form partial sums are unavailable
Infinite Series — Converges when its sequence of partial sums has a finite limit
Geometric Series — A classic series with a known partial sum formula
Telescoping Series — Series where partial sums simplify by cancellation