Examples of Infinite Sequences including the Fibonacci Sequence, from Edwards & Penney

presentation of Edwards & Penney's work, by by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

In the following example of infinite sequences, there are three notations for the same series. The first is a concise notitation of the form {an}, the second is in the function form for the nth term, and the third is the extended eliptical list notation.

Here we see that the sequence begins with n being substituted by 1, followed by the sequence of integers, to infinity.

Another example of an infinite series is the Fibonacci Sequence:

From this formula, we see that the first two terms are set as 1 and 2. Then, every term after 2 obtains its value by summing the previous two terms, resulting in:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . .

The Fibonacci series is an example of a recursively defined sequence: each of its terms (after the first 2) is given by a formula involving its predecessors.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002,p683a,c.