Finite Series¶

Definition¶

Let $a_1,\ldots,a_n$ be real numbers. Set $S_0=0$ and, for $k=1,\ldots,n\text{,}$ define

S_k=S_{k-1}+a_k.

The finite series $\displaystyle\sum_{i=1}^n a_i$ is $S_n\text{,}$ so that $\displaystyle\sum_{i=1}^n a_i=a_1+\cdots+a_n\text{.}$ The empty sum is $\displaystyle\sum_{i=1}^0 a_i=0\text{.}$

Interpretation¶

A finite series represents the sum of the terms of a finite sequence.

The recursion adds one term at a time: the partial sum $S_k$ is the running total after the first $k$ terms, so $S_n$ is the total of all $n$ terms. Starting from $S_0=0$ means that a sum with no terms is $0$ .

The index is only a label: $\displaystyle \sum_{i=1}^n a_i$ and $\displaystyle \sum_{k=1}^n a_k$ are the same number, and so is the shifted form $\displaystyle \sum_{i=0}^{n-1}a_{i+1}$ . Constants can be taken out and sums can be split: $\displaystyle \sum_{i=1}^n(\alpha a_i+\beta b_i)=\alpha\sum_{i=1}^n a_i+\beta\sum_{i=1}^n b_i$ $β i = 1 \sum n b_{i}$ . A common slip is to treat sums like products: $\displaystyle \sum a_ib_i$ is in general not $\displaystyle \bigl(\sum a_i\bigr)\bigl(\sum b_i\bigr)$ ; with $a_i=b_i=1$ for $i=1,2$ $1, 2$ , the first is $2$ and the second is $4$ . Similarly, $\displaystyle \sum_{i=1}^n c=nc$ $n c$ , not $c$ .

Infinite series are built from these partial sums, and the arithmetic and geometric series formulas are closed forms for $S_n$ .

Key Results¶

Every finite series has a well-defined sum, obtained by the recursion above.
The empty sum equals $0\text{,}$ which keeps the recursion valid at $k=1\text{.}$
Finite series do not involve convergence issues.

Examples¶

For $a_i=i$ $i$ , the recursion gives $S_1=1$ $1$ , $S_2=1+2=3$ $3$ , $S_3=3+3=6$ $6$ , $S_4=6+4=10$ and $S_5=10+5=15$ $15$ , so $\displaystyle \sum_{i=1}^5 i = 15\text{.}$
$\displaystyle\sum_{i=1}^3 2^i = 2+4+8=14\text{.}$

Prerequisites¶

Finite Sequence

Sources¶

Math I: Calculus · Lecture 10

Aliases: finite sum Symbols: alpha, beta, sum

Last updated: 20 October 2026 03:02