Series Concept

Adding the Terms of a Sequence

How can the terms of a number sequence be accumulated? An expression such as $1+2+3+\cdots$ is called a series.

A number series is formed by adding the terms of a number sequence in their given order. Two common examples are an arithmetic series, whose terms come from an arithmetic sequence, and a geometric series, whose terms come from a geometric sequence.

In the handshake example below, each new person adds another term to the sum.

Partial Sums Connect a Sequence to Its Series

For a sequence $(U_n)$, the sum of its first $n$ terms is the $n$th partial sum:

Finite problems ask for a particular partial sum. For an infinite series, we study whether the sequence $(S_n)$ approaches a finite limit as $n$ grows.

Number of Handshakes

Imagine there are several people in a group. If each person shakes hands exactly once with every other person in the group, how many total handshakes occur?

Count the handshakes step by step:

• If there are $2 \text{ people}$: There is only $1 \text{ handshake}$.
• If there are $3 \text{ people}$: The first person shakes hands with $2 \text{ people}$. The second person has already shaken hands with the first, so they only need to shake hands with the third person. The total is $2 + 1 = 3 \text{ handshakes}$.
• If there are $4 \text{ people}$: Following the same pattern, we get $3 + 2 + 1 = 6 \text{ handshakes}$.
• If there are $5 \text{ people}$: The total is $4 + 3 + 2 + 1 = 10 \text{ handshakes}$.

The number of handshakes forms a sequential sum of natural numbers.

Summations like $1+2+3+4$ are examples of a number series. In this case, the series is formed from the sum of the terms of the sequence of natural numbers $(1, 2, 3, 4, ...)$.

Each pair must be counted exactly once. Counting every person's handshakes independently would count each pair twice, once from each person's point of view.

A Second Counting Method

A handshake is an unordered pair of people. The number of pairs chosen from $n$ people is:

For $n=5$, this gives $\binom52=10$, matching the sequential sum.

Arithmetic and Geometric Patterns

The handshake totals $(1, 3, 6, 10, ...)$ are neither arithmetic nor geometric because consecutive terms have no constant difference or ratio ($3-1=2$, $6-3=3$, $10-6=4$).

Each expression $1$, $1+2$, $1+2+3$, and $1+2+3+4$ is a partial sum of the natural-number sequence. These partial sums show how a series is built from the terms of a sequence.

Practice

How many handshakes occur among $12 \text{ people}$ if every pair shakes hands exactly once?

The sequential-sum method gives:

The pair-counting formula gives the same result directly:

There are $66 \text{ handshakes}$.