MAT 1348 Study Guide - Final Guide: Antiderivative
14 Oct 2019
School
Department
Course
Professor
MAT1322E CALCULUS II ELIZABETH MALTAIS
5. Sequences
⇧The importance of sequences and series in calculus stems from Newton’s idea of
representing a function as the sum of an infinite series.
⇧For example, for a function like e−x2, we don’t have an explicit algebraic formula for its
antiderivative, but we can rewrite e−x2as the sum of an infinite series for which we can
integrate each term.
⇧Physicists also use series to analyze phenomena by replacing a “complicated” function
by the first few terms in the series that represents it.
SEQUENCES
•Asequence can be thought of as an ordered list of numbers:
•The number a1is the first term. The number a2is the second term. The number anis the
nth term, and so on.
•Each number anin the sequence has a successor an+1 so that the sequence is infinite.
•In fact, a sequence is a special type of function!
∗These notes are solely for the personal use of students registered in MAT1322.
1
Al ,Az ,A},a oo sAn sAnti ,...
•for every positive integer n>I,we can think of the term an of the
sequence as the value far )of afunction fwhose domain is 'Z+ and
whose codomainis LR
,
E± f(n )=#⇒f(1) =tt ,
=If(2) =#=23 ft )= 5+4=34
Thus ,f- defines the sequence ftp.FC2 ),f(3) ,
...
tz sZz ,},4g ,§,. a
Notation .We Write a,.az )azsn .instead of a(1) ,a(2) ,a(3) ,
...
•we also write a,sazsazsn .more compactly as {an }or {an }F=,
E± .{he
,}denotes the sequence ÷,}s¥s in
