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Unformatted text preview: Math 191 Problem Set for Week 12 Section Definition 0.1 (Infinite Sequences) An infinite sequence of numbers is a function whose domain is the set of positive integers. Ex 1. Write out the first few terms of the sequences a1 = 1, an+1 = an + Ex 2. Find a formula for the nth term of the sequence. a. The sequence 1,  4, 9,  16, 25, ... b. The sequence 1, 0, 1, 0, 1, ... Theorem 0.2 Suppose that f (x) is a function defined for all x n0 and that {an } is a sequence of real numbers such that an = f (n) for n n0 . Then
x 1 2n . lim f (x) = L lim an = L
x The significance of Theorem 0.2 is to enable us to use L'H^pital's Rule to find the limits of some sequences. o Ex 3. Find lim an when an =
n 15n4 n4 +8n3 . (ln n)200 . n Ex 4. Find lim an when an =
n Definition 0.3 (Geometric Series) a. If r < 1,
n=1 arn1 = a . 1r b. If r 1,
n=1 arn1 = . Diverges. Ex 5. Find 7 . 4n n=1 Ex 6. Find
n=0 1 (1)n + . n 2 5n Theorem 0.4 a. If
n=1 an converges, then an 0. Note that the converse is not true, i.e. an 0
1 n. an converges. For example, an =
n=1 b. If an fails to exist or is difference from zero, then
n=1 an diverges. Ex 7. Does
n=0 1 2 n converge or diverge? If a series converge, find its sum. Ex 8. Does (1)n+1 n converge or diverge? If a series converge, find its sum.
n=0 1 ...
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This note was uploaded on 06/20/2008 for the course MATH 1910 taught by Professor Berman during the Spring '07 term at Cornell University (Engineering School).
 Spring '07
 BERMAN
 Math, Calculus, Integers

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