notes Overview of Sequences and Series - Overview of Sequences and Series MAT 104 Frank Swenton Summer 2000 Sequences are ordered lists of real numbers

Thomas' Calculus: Early Transcendentals

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Overview of Sequences and Series MAT 104 – Frank Swenton, Summer 2000 Sequences are ordered lists of real numbers, such as “ a 1 , a 2 , a 3 , . . . ”, sometimes written { a n } Just as with limits of functions on the real line, we can talk about limits of sequences; if the numbers in the sequence approach some real number L , then the sequence has a limit. Otherwise, it may approach or -∞ , or it may not approach anything at all in particular. Note that if lim x →∞ f ( x ) either exists or is infinite, then the limit lim n →∞ f ( n ) is the same (the graph of the sequence lies on the graph of the function, so if the function approaches a limit, the sequence is stuck to the same limit). This fact can be useful in computing limits of some sequences, since limits of functions can often be more easily evaluated by l’Hˆ opital’s rule. For example: Using l’Hˆ opital’s rule, lim x →∞ x 2 e x = lim x →∞ 2 x e x = lim x →∞ 2 e x = 0, so lim n →∞ n 2 e n = 0 as well. Some useful limits to know: If | r | < 1, then lim n →∞ r n = 0 lim n →∞ n n = 1 (and the same for n n 2 , n n 3 , etc.) lim n →∞ n n ! = Some limits that equal e : lim n →∞ 1 + 1 n n , lim n →∞ n + 1 n n Some limits that equal 1 e : lim n →∞ 1 - 1 n n , lim n →∞ n - 1 n n , lim n →∞ n n + 1 n Remember the hierarchy of functions (on the “limit comparison test” page); If f ( n ) g ( n ), then lim n →∞ f ( n ) g ( n ) = 0, and lim n →∞ g ( n ) f ( n ) = . For example, lim n →∞ 100 n n ! = 0, lim n →∞ n 100 2 n = 0, etc. Remember the rules of exponentiation: a m + n = a m · a n , a - n = 1 a n , and a mn = ( a m ) n Series (sums of infinitely many terms) Given a sequence a 1 , a 2 , a 3 , . . . , we may want to add up all of the values, i.e. a 1 + a 2 + a 3 + · · · . This is called a series , and it is denoted X n =1 a n . The individual numbers being added together are called the terms of the series. To add up an infinite number of terms, we first define the partial sums of the series, S n = a 1 + a 2 + · · · + a n , i.e. S n is the sum of the first n terms of the series. We then define the meaning of the infinite sum by X n =1 a n = lim n →∞ S n . If this limit exists as a real number, we call the series convergent ; if the limit doesn’t exist, we call the series divergent .