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JH
Numerade Educator

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Problem 86 Medium Difficulty

If $ \sum a_n $ and $ \sum b_n $ are both divergent, is $ \sum \left(a_n + b_n \right) $ necessarily divergent?

Answer

$\mathrm{No}$

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Video Transcript

if the sum of a n and the sum of being or both diversion is the sum of a M plus being necessarily diversion? Well, not necessarily. So here. Not necessarily. Sometimes it is. Sometimes it converges. Not necessarily. So hear my example. Let's take a N to be two and let's take being to be minus two. Then we have a n equals to this diverges No, by the divergence test. So if you take here the limit of a N, it's just a limit of two, which equals two, and it's not equals zero and anytime the limit of and is non zero, that means it diverges. Similarly, this bee in Siri's here also diverges by the diversions test. So we have two diversion Siri's and in B end. However, what happens when we look at the sum of a and plus being well, if you just write this out am is too bn is negative too. So we just get an infinite sum of zeros and it doesn't matter how many zeros we add. Since all of these numbers are zero, the entire sum will be zero and that before it converges, so it is possible for the sum of and to be and to be convergent so that it's not necessarily diversion, and that's our final answer.