Convergence of Sequences 2.3 2002/10/03 2003/04/15 Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu Richard G. Baraniuk richb@rice.edu convergence pointwise point wise sequences norm convergence norm This module will present an introduction into convergence and focus on what a sequence is and how it behaves as it approaches infinity.

What is a Sequence? sequence A sequence is a function gn defined on the positive integers 'n'. We often denote a sequence by n 1 gn A real number sequence: gn 1 n A vector sequence: gn n 2 n 2 A function sequence: g n t 1 0 t 1 n 0 A function can be thought of as an infinite dimensional vector where for each value of 't' we have one dimension

Convergence of Real Sequences limit A sequence n 1 gn converges to a limit g if for every ε 0 there is an integer N such that i i N gi g ε We usually denote a limit by writing i gi g or g i g The above definition means that no matter how small we make ε, except for a finite number of g i 's, all points of the sequence are within distance ε of g. We are given the following convergent sequence: gn 1 n Intuitively we can assume the following limit: n gn 0 Let us prove this rigorously. Say that we are given a real number ε 0 . Let us choose N 1 ε , where x denotes the smallest integer larger than x. Then for n N we have gn 0 1 n 1 N ε Thus, n gn 0 Now let us look at the following non-convergent sequence gn 1 n even -1 n odd This sequence oscillates between 1 and -1, so it will therefore never converge.

Problems For practice, say which of the following sequences converge and give their limits if they exist. gn n gn 1 n n even -1 n n odd gn 1 n n power of 10 1 gn n n 10 5 1 n n 10 5 gn n gn n