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May 30th, 2017, 06:46 AM   #1
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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
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May 31st, 2017, 04:07 AM   #2
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Quote:
Originally Posted by Lalitha183 View Post
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
Please someone check
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May 31st, 2017, 04:24 AM   #3
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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 04:29 AM.
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June 1st, 2017, 04:20 PM   #4
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I have one more doubt. Can you please let me know the difference between Option A and Option C ?
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June 1st, 2017, 04:29 PM   #5
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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 04:40 PM.
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