Sequences+&+Series


 * Series:** a sum of terms
 * Sequence:** a list of terms

__Sigma__ is the greek letter that stands for sum when dealing with series.

The notation looks kind of odd at first, but it's easy to pick up.

An example in summation notation: What the notation means:
 * The stuff below the sigma (usually in the form " i = # " or " n = # ") gives you two pieces of information:
 * 1) The number (3) tells you what integer the "changing variable" starts at
 * 2) The letter (i) tells you what variable changes at each step
 * The number above the sigma (7) tells you what the "changing variable" stops at
 * The expression on the right of the sigma (i^2) tells you what gets added to the final sum at each step

So, here's what this particular expression means:

If you understand completely how it works by looking at the solution above, great!

If not, here's the breakdown:
 * 1) It takes the initial value from the bottom (3), plugs it in for i, and then gives you what the expression to the right of the sigma is when i = 3 (i ^ 2 = 3 ^ 2 = 9)
 * 2) Then, it adds one to i (making it 4), and plugs it in to the expression again to give you i ^ 2 = 4 ^ 2 = 16. At this point, the running sum is 9 + 16.
 * 3) It repeats this over and over for every value of i from 3 to 7
 * 4) Finally, //after// it computes i ^ 2 for i = 7 (don't stop at 6), it adds it to the running sum and gives you the final sum.

For the computer-programming-inclined mathlete, this is kind of like using a for-loop to add stuff to a variable For the non-computer-programming-inclined mathlete, don't worry about it :-)


 * __Special types of series:__**


 * Arithmetic Series:** has a common difference

Example: 1 + 2 + 3 + 4 + 5 (common difference = 1) Example: 2 + 6 + 10 + 14 (common difference = 4) Example: 1 + 0.8 + 0.6 + 0.4 + 0.2 + 0 - 0.2 -0.4 (common difference = -0.2)


 * You can only find the sum of a finite number of terms in a series

Formula


 * Geometric Series**: has a common multiple

Example: 1 + 2 + 4 + 8 + 16 (common multiple = 2) Example: 3 - 9 + 27 - 81 (common multiple = -3) Example: i + i^2 + i^3 + i^4 (common multiple = i) Example: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... (common multiple = 1/2; can be solved using infinite sum formula below)
 * [[image:sum_of_a_geometric_series.gif]] where **a** is the first term and **r** is the common ratio (only for //infinite// sums adding up all terms from the 1st to infinity)
 * if [[image:r_is_greater_than_1.gif]] there is no sum (try adding 1 + 2 + 4 + 8 + ... to infinity using this formula - it spits out a negative in frustration)

__**More sequence stuff:**__

__Capital Pi__ above is the greek letter that stands for "product" when dealing with sequences.

Essentially, it's the exact same thing as sigma, except you multiply instead of adding. Even the notation is the same!

For example, Means "1 * 2 * 3 *4 * ... * 10"

Another example is shown below:

This sometimes comes up in logarithm and imaginary number problems, but it could show up just about anywhere.