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1. Let {a_n} be a nonincreasing sequence of nonnegative numbers such that sum(a_k, k=1..+oo) <+oo. Let S denote the set of all numbers which are sums of subseries of sum(a_k,k=1..+oo). Prove that S is an interval if and only if for each n in N*, a_n <= sum(a_k,k=n+1..+oo).
Solution


2. Let (a_n) be a sequence of positive real numbers. Prove that: sum(a_n, n=1..+oo) = +oo -> sum(a_n/(1 + a_n),n=1..+oo) = +oo.
Solution


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