I am trying to make sense of the comparison test and I follow the proof for the comparison test for the convergence then the author appears to use the contra-positive to show if you flip the statements around and show that one diverges then the other must.

But that only makes sense to me if I assume that if a series does not converge then it must diverge . Is this true? Thanx.


closed as off-topic by Andrés E. Caicedo, Rohan, José Carlos Santos, Namaste, user223391 Dec 28 '17 at 2:40

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    $\begingroup$ It is true by definition. $\endgroup$ – Andrés E. Caicedo Dec 23 '17 at 16:03
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    $\begingroup$ People use "diverge" in two related but fundamentally different ways. I believe that standard usage is to say that any series which does not converge must diverge. However, people often distinguish between a series for which the partial sums go to $\infty$ and a series for which the partial sums oscillate, as in $\sum (-1)^n$. You should clarify what you mean by "diverges". $\endgroup$ – lulu Dec 23 '17 at 16:03
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    $\begingroup$ You find definitions in textbooks. Thank you. $\endgroup$ – Professor Vector Dec 23 '17 at 16:04
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    $\begingroup$ I've always taken "diverges" to mean "does not converge". There's also a concept of "diverging to" something, such as $+\infty$ or $-\infty$ or (in the case of infinite products) $0. \qquad$ $\endgroup$ – Michael Hardy Dec 23 '17 at 16:59
  • $\begingroup$ your comments have answered the question.....it is by definition and convention ....thank you $\endgroup$ – Sedumjoy Dec 24 '17 at 16:35