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Prove that if the the terms of a sequence decrease monotonically, $a_1 \geq a_2 \geq a_3 \geq \dots \geq a_n$, and converge to 0 then the series $\Sigma a_k$ converges iff the associated dyadic series $a_1 + 2a_2 + 4a_4 + \dots + = \Sigma 2^k a_{2^k}$ converges.

Given: $a_1 \geq a_2 \geq a_3 \geq \dots \geq a_n$, and converges to 0. \ \textbf{Part 1:} \ {Assume} $a_1 + 2a_2 + 4a_4 + \dots + = \Sigma 2^k a_{2^k}$ is convergent. Then, we have: \ $a_1 \leq a_1$ \ $a_2 + a_3 \leq a_2 + a_2 = 2a_2$ \ $a_4 + a_5 + a_6 + a_7 \leq a_4 + a_4 + a_4 + a_4 = 4a_4$ \ $a_8 + \dots + a_{15} \leq 8a_8$ \ \noindent Adding the inequalities above, we have: \ $\Sigma_{n=1}^\infty a_n \leq \Sigma_{k=0}^\infty 2^k a_{2^k}$ \ Thus, since the RHS is convergent and all the terms are positive, by the Comparison Test, the LHS must also be convergent. Hence, $\Sigma_{n=1}^\infty a_n$ is convergent. \ \textbf{Part 2:} \ {Assume} $\Sigma_{n=1}^\infty a_n$ converges. Then, $a_1 + 2 \Sigma_{n=1}^\infty a_n$ also converges since we are merely scaling by a constant and adding a constant term. \ Recall that $a_1 \geq a_2 \geq a_3 \geq \dots \geq a_n$ Then, \ $a_1 \leq a_1$ \ $2a_2 \leq 2a_1$ \ $4a_4 \leq 2a_2 + 2a_3$\ $8a_4 \leq 2a_4 + 2a_5 + 2a_6 + 2a_7$\ Adding the inequalities above, we have: \ $\Sigma_{k=0}^\infty 2^k a_{2^k} \leq a_1 + 2 \Sigma_{n=1}^\infty a_n$\ Since the RHS converges and all elements are positive, the LHS must converge by the Comparison Test. Hence, $\Sigma_{k=0}^\infty 2^k a_{2^k}$ converges.

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  • $\begingroup$ Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. $\endgroup$ – Clement C. Nov 10 '17 at 22:37
  • $\begingroup$ Possible duplicate of: math.stackexchange.com/questions/1782023/… $\endgroup$ – Ben P. Nov 10 '17 at 22:38
  • $\begingroup$ A tip: don't use the tag "proof-verification" if you do not provide a proof to be verified. Another tip: the keywords are Cauchy condensation test. $\endgroup$ – Clement C. Nov 10 '17 at 22:38
  • $\begingroup$ Possible duplicate of Proving Cauchy condensation test $\endgroup$ – Clement C. Nov 10 '17 at 22:42
  • $\begingroup$ Sorry I did add a proof. Does it look okay though? $\endgroup$ – nrc123 Nov 10 '17 at 22:43

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