Series

**Series**

When a function is only defined for the natural numbers (1, 2, 3, 4, 5, ...), we can call the set of function values a **sequence**. For example, the function *a _{n}* = 3

*n*- 2 leads to the sequence {1, 4, 7, 10, 13, ...} when

*n*takes on the value of each successive natural number.

A **series** is the value you get when you add up all the terms of a sequence. This value is called the sum. It is denoted by the capital Greek letter, sigma: ∑

For example, to show the summation of the first ten terms of a sequence {*a _{n}*}, the symbol notation would be as follows:

where *n* = 1 is the lower index, corresponding to the first term, and 10 is the upper index, corresponding to the last term. The *a _{n}* stands for the terms to be added. The notation is described as "the sum as

*n*goes from one to ten of

*a*-sub-

*n*" and can be expanded as follows:

a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10}

**arithmetic series**, a series based on an arithmetic sequence, you can find the sum quickly by using this formula:

*f(*1) and

*f*(

*N*) represent the first and last terms respectively.

**Example:**Find the sum 1 + 3 + 5 + 7 + ... + 99.

^{50}/

_{2}(1 + 99) = 25(100) =

**2500**.

**geometric series**, a series based on a geometric sequence, you can find the sum by using this formula:

**Example:**Find the sum 3 + 6 + 12 + 24 + ... + 1536.

*a*is the first term in the series, which is 3. If you can't figure out right away how many terms there are in the series, do some reverse-engineering on the last term:

*ar*

^{ n}^{-1}= 1536

^{n-1}) = 1536

^{n-1}= 512

^{n-1}= 2

^{9}

*n*- 1 = 9

*n*= 10

*N*= 10, we can use the formula:

^{10}) ÷ (1 - 2) =

**3069**

*r*that satisfies |

*r*| < 1, you can find the sum of the entire series:

**Example:**Find the sum 50 + 10 + 2 +

^{2}/

_{5}+ ...

^{1}/

_{5}. Because the common ratio causes the numbers to get smaller and smaller, the sum is not infinite. Using the common ratio of

^{1}/

_{5}and the first term of

*a*= 50, we can find the sum:

^{1}/

_{5}) = 50 ÷

^{4}/

_{5}=

^{250}/

_{4}=

**62.5**