Let Ea(n) be such that (a(n)) is a decreasing sequence of strictly positive numbers. If s(n)...

 

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Let Ea(n) be such that (a(n)) is a decreasing sequence of strictly positive numbers. If s(n) n=1 denotes the nth partial sum, show (by grouping the terms in s(2") in two different ways) that (a(1) + 2a(2) +..+2"a(2")) < s(2") < (a(1) + 2a(2) + ...+ 2"-1a(2"-1)) + a(2"). Use these inequalities to show that E a(n) converges if and only if 2" a(2") converges. This n=1 n=1 result is often called the Cauchy Condensation Test; it is very powerful. Let Ea(n) be such that (a(n)) is a decreasing sequence of strictly positive numbers. If s(n) n=1 denotes the nth partial sum, show (by grouping the terms in s(2") in two different ways) that (a(1) + 2a(2) +..+2"a(2")) < s(2") < (a(1) + 2a(2) + ...+ 2"-1a(2"-1)) + a(2"). Use these inequalities to show that E a(n) converges if and only if 2" a(2") converges. This n=1 n=1 result is often called the Cauchy Condensation Test; it is very powerful.

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