Problem 46 Medium Difficulty

Find all values of $ c $ for which the following series converges.
$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {c}{n} - \frac {1}{n + 1} \right) $

Answer

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Video Transcript

we'd like to know the values of sea that allowed the Siri's to converge. So here one option is to go ahead and try the integral test. And other option is they're just proceed out your break. Leah's follows. So here, let me go ahead and just rewrite this C A. C minus one plus one and then I'll split ups in terms here. C minus ones constant so I can pull that out. Notice that this first Siri's over here. This's a telescoping Siri's, and this can be evaluated, and we will actually see that the Siri's converges. So let's just focus on this series for a second. And let's rewrite it as a limit. Kay goes to infinity. Now let's go ahead and evaluate that partial son. Okay? We would keep going all the way until you plugged in less maybe do a few terms before okay was through K minus two and then came on this one and then kay. And now we should cancel is much as we can. So we see the first term we'LL stay there the one But all these intermediate terms cancel out all, even all the way until you get to came on this one. Okay, However, this one over came on. This one will also stay. So you just have the one in the final term. That's day. And when you take the limit, this fraction or this expression goes toe one because the fraction goes to zero. So that shows that this summit is just equal to one so we can rewrite the whole expression as one plus C minus one someone over end. However, if you look at the other Siri's here, our circle is in black. This's harmonic, or you can say it. It's P Siri's with people's one, So this diverges. So unless we want this Siri's to diverge, we should go ahead and take C equals one, because then they'LL wipe off this here, and that will ensure that our some conversions to just the number one so converges. If and only if C equals one