May 30th, 2017, 07:46 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Sequences
Consider the real sequence $(a_n)$ and $(b_n)$ such that $\displaystyle \sum a_n b_n $ converges. Which of the following statements is true ? A) If $\displaystyle \sum a_n $ converges, then $(b_n)$ is bounded. B) If $\displaystyle \sum b_n $ converges, then $(a_n)$ is bounded. C) If $(a_n)$ is bounded, then $(b_n)$ converges. D) If $(a_n)$ is unbounded, then $(b_n)$ bounded. I'm guessing it as option A. Please check and let me know Thank you 
May 31st, 2017, 05:07 AM  #2  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
 
May 31st, 2017, 05:24 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra 
If A is true, then B must also be true. I think, from memory, that both are. If $(a_n)$ is bounded, but not convergent, then $(b_n)$ must converge to zero. This is a necessary condition but not sufficient. Similarly, if $(a_n)$ is unbounded, then $(b_n)$ must converge to zero. This is a necessary condition but not sufficient. Last edited by v8archie; May 31st, 2017 at 05:29 AM. 
June 1st, 2017, 05:20 PM  #4 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 
I have one more doubt. Can you please let me know the difference between Option A and Option C ?

June 1st, 2017, 05:29 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra 
In option C, the sum $\sum a_n$ doesn't necessarily converge. In fact, the sequence $(a_n)$ doesn't even have to converge $a_n = \sin n$ would be one such sequence. Equally, although the sequence $b_n$ converges, the series $\sum b_n$ doesn't necessarily converge even if $(b_n)$ converges to zero. e.g. $b_n = \frac1n$. Looking again at the question, given that $\sum a_n b_n$ converges then A and B can be false. For example: $a_n = \frac1{n^3},\, b_n = n$. C can also be false using the same example. D is true, because if the sequence $(a_n)$ is not bounded, the series $\sum b_n$ must converge quickly enough to make the sum $\sum a_n b_n$ converge. An example of this is $a_n=n,\, b_n=\frac1{n^3}$. Last edited by v8archie; June 1st, 2017 at 05:40 PM. 

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