Copy of s6 - Solutions to homework 6 1 If either lim sup on...

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Solutions to homework # 6. 1. If either lim sup on the right-hand side is + , then the inequality is trivially satisfied. Also, if lim sup n →∞ a n = -∞ , then ( a n ) tends to -∞ ; if lim sup n →∞ b n < , then the sequence a n + b n tends to -∞ as well. So, it remains to consider the case when both lim sup a n and lim sup b n are finite. Take any subsequence ( a n k + b n k ) that tends to lim sup n →∞ ( a n + b n ). Since ( a n k ) is a subsequence of ( a n ), we conclude that lim sup k →∞ a n k lim sup n →∞ a n . Let ( a n k l ) be the subsequence of a n k that tends to lim sup k →∞ a n k . Since lim sup l →∞ b n k l lim sup n →∞ b n , we obtain lim sup( a n + b n ) = lim l →∞ ( a n k l + b n k l ) lim sup a n + lim sup b n . 2. (a) Since a n = 1 / ( n + 1+ n ) > 1 / (3 n ) and the series n 1 / n diverges, we conclude by the comparison test that the series a n diverges as well. (b) Since
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