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Unformatted text preview: s if and only if every subsequence has a subsequence which converges to s . Students registered for 18.100C should write this problem up in LaTeX. 4. Problem 24 from page 82. Part 3 5. Problem 6 from page 78. 1 6. Let ( a n ) n ∈ N be a sequence of positive numbers which tends to zero but such that ∑ ∞ n =1 a n diverges. Let ( A n ) n ∈ N be the sequence of partial sums A n = n X k =1 a k , and let b n +1 = √ A n +1-√ A n . Show that lim n →∞ b n a n = 0 , but that ∑ ∞ n =1 b n is still divergent. In this sense there is no ‘smallest’ divergent series, and one can similarly show that there is no ‘largest’ convergent one. 2...
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- Fall '10
- Limit of a sequence, lim sup, lim sup bn