Sequences and Series: Terminology and Notation

The sequence can also be written in terms of its terms. For instance, the sequence of terms a_i, with the index running from i = 1 to i = n, can be written as:

The sequence of terms starting with index 3 and going on forever could be written as:

Some books use the parenthesis notation; others use the curly-brace notation. Either way, they're talking about lists of terms. The beginning value of the counter is called the "lower index"; the ending value is called the "upper index". The formatting follows the English: the lower index is written below the upper index, as shown above. (The plural of "index" is "indices", pronounced INN-duh-seez.)

Note: Sometimes sequences start with an index of n = 0, so the first term is actually a₀. Then the second term would be a₁. The first listed term in such a case would be called the "zero-eth" term. This method of numbering the terms is used, for example, in Javascript arrays. Or, as in the second example above, the sequence may start with an index value greater than 1. Don't assume that every sequence and series will start with an index of n = 1.

When a sequence has no fixed numerical upper index, but instead "goes to infinity" ("infinity" being denoted by that sideways-eight symbol, ∞), the sequence is said to be an "infinite" sequence. Infinite sequences customarily have finite lower indices. That is, they'll start at some finite counter, like i = 1.

As mentioned above, a sequence A with terms a_n may also be referred to as "{a_n}", but contrary to what you may have learned in other contexts, this "set" is actually an ordered list, not an unordered collection of elements. (Your book may use some notation other than what I'm showing here. Unfortunately, notation doesn't yet seem to have been entirely standardized for this topic. Just try always to make sure, whatever resource you're using, that you are clear on the definitions of that resource's terms and symbols.) In a set, there is no particular order to the elements, and repeated elements are usually discarded as pointless duplicates. Thus, the following set:

...would reduce to (and is equivalent to):

On the other hand, the following sequence:

...cannot be rearranged or "simplified" in any manner.

The terms of a sequence can be simply listed out, as shown above, or else they can be defined by a rule. Often this rule is related to the index. For instance, in the sequence A = {a_i} = {2i + 1}, the i-th term is defined by the rule "2i + 1", so the first few terms are:

...and so forth. Sometimes the rule for a sequence is such that the next term in the sequence is defined in terms of the previous terms. This type of sequence is called a "recursive" sequence, and the rule is called a "recursion". The most famous recursive sequence is the Fibonacci (fibb-oh-NAH-chee) sequence. Its recursion rule is as follows:

What this rule says is that the first two terms of the sequence are both equal to 1; then every term after the first two is found by adding the previous two terms. So the third term, a₃, is found by adding a₃₋₁ = a₂ and a₃₋₂ = a₁. The first few terms of the Fibonacci sequence are:

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To indicate a series, we use either the Latin capital letter "S" or else the Greek letter corresponding to the capital "S", which is called "sigma" (SIGG-muh):

To show the summation of, say, the first through tenth terms of a sequence {a_n}, we would write the following:

Just as with the terminology for sequences, the "n = 1" is called the "lower index", telling us that "n" is the counter and that the counter starts at "1"; the "10" is called the "upper index", telling us that a₁₀ will be the last term added in this series; "a_n" stands for the terms that we'll be adding. The whole thing is pronounced as "the sum, from n equals one to ten, of a-sub-n". The summation symbol above means the following:

The written-out form above is called the "expanded" form of the series, in contrast with the more compact "sigma" notation.

Any letter can be used for the index, but i, j, k, m, and n are probably used more than any other letters.

There are some rules that can help simplify or evaluate series. If every term in a series is multiplied by the same value, you can factor this value out of the series. This means the following:

This means that, if you've been told that the sum of some particular series has a value of, say, 15, and that every term in the series is multiplied by, say, 2, you can find the value as:

The other rule for series is that, if the terms of the series are sums, then you can split the series of sums into a sum of series. In other words:

If you add up just the first few terms of a series, rather than all (possibly infinitely-many) of them, this is called "taking (or finding) the partial sum". If, say, you were told to find the sum of just the first eight terms of a series, you would be "finding the eighth partial sum".

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Sequences and series are most useful when there is a formula for their terms. For instance, if the formula for the terms a_n of a sequence is defined as "a_n = 2n + 3", then you can find the value of any term by plugging the value of n into the formula. For instance, a₈ = 2(8) + 3 = 16 + 3 = 19. In words, "a_n = 2n + 3" can be read as "the n-th term is given by two-enn plus three". The word "n-th" is pronounced "ENN-eth", and just means "the generic term a_n, where I haven't yet specified the value of n."

Of course, there doesn't have to be a formula for the n-th term of a sequence. The values of the terms can be utterly random, having no relationship between n and the value of a_n. But sequences with random terms are hard to work with and are less useful in general, so you're not likely to see many of them in your classes.