4802Series07 - P be the set of all positive integers of the...

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Math 4802 Fall 2007 Infinite Series Problems (due Tuesday, October 23) 1. Evaluate the infinite sum X n =1 n 2 2 n . 2. Suppose that a sequence a 1 , a 2 , a 3 , . . . of positive real numbers satisfies a n a 2 n + a 2 n +1 for all n 1. Prove that the series n =1 a n diverges. 3. Let N be the set of all positive integers which do not contain the digit 9 in their decimal representation. Prove that X a N 1 a < 80 . 4. Let
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Unformatted text preview: P be the set of all positive integers of the form a b with a, b 2 integers. Prove that X q P 1 q-1 = 1 . 5. Prove that if n =1 a n is a convergent series of positive real numbers, then so is n =1 ( a n ) n/ ( n +1) . 6. Show that Z 1 1 x x dx = X n =1 1 n n (where by convention we set 0 = 1). 1...
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