ps6 - a n +1 a n + 1 n 2 , n N prove that ( a n ) n N...

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18.100B and 18.100C Fall 2011 Problem Set 6 Due October 27th at 4 pm in room 2-108. Hand in parts 1, 2 and 3 separately. Put your name and whether you are registered for 18.100B or 18.100C on each part. Part 1 1. Problem 9 from page 79. 2. Suppose ( a n ) n N satisfies a n + m a n + a m for all m,n N . Prove that lim n →∞ a n n = inf n a n n : n N o , as an element of R ∪ {-∞} . You may use without proof the Euclidean division algorithm , which says that for any n,‘ N , there exist unique m,r N ∪ { 0 } with r < ‘ such that n = m‘ + r. Part 2 3. If ( a n ) n N is a sequence of nonnegative real numbers satisfying
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Unformatted text preview: a n +1 a n + 1 n 2 , n N prove that ( a n ) n N converges. 4. Prove that if ( a n ) n N is a sequence of real numbers such that | a n +1-a n | converges, then ( a n ) n N converges. Prove that the converse does not hold. 5. Prove that ( a n ) n N converges if and only if (2 a n +1-a n ) n N does. Part 3 6. Problem 4 from page 98. Students registered for 18.100C should write this problem up in LaTeX. 7. Problem 23 from page 101. 1...
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This note was uploaded on 02/15/2012 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

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