Math191_Week13_section_without_sol

Math191_Week13_secti - Math 191 Problem Set for Week 12 Section Definition 0.1(Infinite Sequences An infinite sequence of numbers is a function

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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 1-5n4 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 arn-1 = a . 1-r b. If |r| 1, n=1 arn-1 = . 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).

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