If Σa_n is divergent, the absolute value of Σa

May 2021 2 267 Report

If Σa_n is divergent, the absolute value of Σa_n is divergent. T/F?

If Σa_n is divergent, the absolute value of Σa_n is divergent. True or false? I fell that it would still be divergent. But I am not sure as to why.

Update:

But Σa_n is divergent not convergent to begin with. So what type of divergent series is convergent when the absolute value is taken?

Science & Mathematics / Mathematics

KevinM

Verified answer

The posting above answered the wrong question - and I think you miswrote your question here. I think you meant:

"If Σa_n is divergent, the Σ |a_n| is divergent. T/F?"

The question as asked is trivial - if a series diverges, the absolute value of that WHOLE SERIES also diverges.

The other question - the one I think you meant - is also true. If a series diverges, then the series of the absolute values of those elements must also diverge.

cakesmckakes

False. Consider the alternating harmonic series, sum([(-1)^k]/k). By the alternating series test it converges. However, taking absolute values of each term gives sum(1/k), which is divergent