### Definition and Notation

There are times when it makes sense to add the numbers in a sequence. The sum of terms of a sequence is called a **series**. A **finite series** is a sum of the terms in a finite sequence.

**Summation notation**is an abbreviated way of writing a sum that uses the Greek letter sigma. This notation is read as "the sum of $a$ sub $n$ as $n$ goes from 1 to 9”:

### Finite Arithmetic Series

An **arithmetic series** is the sum of terms in an arithmetic sequence. The sum can be found by adding the terms in a sequence, but this may become tedious for a large number of terms.

Substitute into the formula for a finite arithmetic series and simplify.

Substitute $n=12$, $a_k=31-3n$, $a_1=28$, and $a_n=-5$.Substitute into the formula for a finite arithmetic series and simplify.

Substitute $n=6$, $a_k=8+2n$, $a_1=10$, and $a_n=20$.### Finite Geometric Series

A **geometric series** is the sum of terms in a geometric sequence. The sum can be found by adding the terms in the sequence, but this may be difficult for a large number of terms.

Substitute into the formula for a finite geometric series and simplify. Round to the nearest hundredth.

Substitute $n=5$, $a_k=40{,}000(1.03)^{n-1}$, $a_1=40{,}000$, and $r=1.03$.### Infinite Series

**infinite series**is the sum of the terms in an infinite sequence. Since an infinite sequence has infinitely many terms, the sum can be found by using partial sums. The sum of a specified number of terms in a sequence is called a

**partial sum**, written as $S_n$.

In an infinite arithmetic series, if the common difference is positive, each term is greater than the previous term, so the sum becomes greater and greater and does not approach a fixed number. It is said to approach $\infty$. If the common difference is negative, each term is less than the previous term, so the sum approaches $-\infty$. When the sequence of partial sums does not approach a fixed number, the series is called a **divergent series**. Since the sum of an infinite arithmetic series does not approach a fixed number, it is undefined.

**convergent series**, which means that the sequence of partial sums approaches that number. The formula for the sum of an infinite geometric series, where $a_1$ is the first term and $r$ is the common ratio, that converges is:

Substitute into the formula for an infinite geometric series with $\left | r \right | <1$ and simplify.

Substitute $a_k=40(0.75)^{n-1}$, $a_1=40$, and $r=0.75$.