Archive for December, 2009

Infinite sequence

An infinite sequence is a function whose domain is the set of positive integers. For example, when n is given in turn the values 1, 2, 3, 4, … the function defined by the formula 1/(n+1) yields the sequence 1/2, 1/3, 1/4, 1/5,… The sequence is called an infinite sequence to indicate that there is no last term.

By the general or nth term of an infinite sequence, we mean a formula s(n) for the value of the function determining the sequence. The infinite sequence itself is often denoted by enclosing the general term in braces, as in {s(n)}, or by displaying hte first few terms of the sequence. For example, the general terms s(n) of the sequence in the preceding paragraph is 1/(n+1), and that sequence can be denoted by {1/(n+1)} or by 1/2, 1/3, 1/4, 1/5,…

Differentiation

A function is said to be differentiable at a point x = x(o) if the derivative of the function exists at that point. Also, a function is said to be differentiable on an interval if it is differentiable at every point of the interval. The functions of elementary calculus are differentiable, except possibly at isolated points, on their intervals of definition. If a function is differentiable, it must be continuous. The process of finding the derivative of a function is called differentiation.

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