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Problem 46 Hard Difficulty

Let $ \sum a_n $ be a series with positive terms and let $ r_n = a_{n+1} / a_n. $ Suppose that $ lim_{n \to \infty} r_n = L < 1, $ so $ \sum a_n $ converges by the Ratio Test. As usual, we let $ R_n $ be the remainder after $ n $ terms, that is,

$ R_n = a_{n+1} + a_{n+2} + a_{n+3} + \cdot \cdot \cdot $

(a) If $ \{r_n\} $ is a decreasing sequence and $ r_{n+1} < 1, $ show, by summing a geometric series, that

$ R_n \le \frac {a_{n+1}}{1 - r_{n+1}} $

(b) If $ \{r_n\} $ is a decreasing sequence, show that


$ R_n \le \frac {a_{n+1}}{1 - L} $

Answer

a. $R_{n} \leq \frac{a_{n+1}}{1-r_{n+1}}$
b. SEE SOLUTION

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Video Transcript

right. I was probably gonna start by setting ourselves equal to a series going for him K equals and plus one to infinity for a So Okay, this is just me rewriting the Siri's. And then based on this definition, I can rewrite Eisuke to be gate to the end close woman times A and close to over a n plus one times a and plus three over a n plus two and so on becoming a k over a K minus one. Oh, and what this allows us to do is rewrite Siri's in terms of, uh, the R series that was given. So then we get hey, and Plus swollen are in plus one are in plus two. So on r k. But this one Now we know that our in is a decreasing Siri's, so we can regard it this whole, um, equation with comparison so that a k it's less than or equal to a and plus one times are the lotus M plus one to the cane, my end minus 21 And so that accounts for all the terms. So now that we know this was to rewrite them bushes mysterious in terms of um, giving ourselves. And that gives us from the vicinity Certain from K equals in plus one te's. Okay, just last center equal Teoh o K equals M plus one to infinity poor a n plus one times are and close Saralyn to the K murders and my sworn And then if you actually plug in the values which will see is that this term is the same thing as if we go from some other arbitrary to recall A J equals zero to a and plus mourn tellings are in plus one to the J. And so with this tells us, is that because harlequins is geometric with the ratio of are in close, one being list holding convergence, we can say that, uh, arson. It's less than or equal to a and plus 1/1 minus r n plus one. And that's just a variation on the geometric formulas a over will notice are. And so because this series converges and this could be this year's Can you run in this form now moving on to part two, we're gonna use the same definition that we have above with this rcep and equal to this pretty well serious, right here and again. The skins is the same definition right here. And then again, we have the same definition. No. Clear. Yeah. So I'm gonna rewrite it in a different formula. Um, go down to the bottom and do that. So here, we're going to stay. That Eisuke is equal to a and plus more are and close one are in plus two. So on giving us RK minus one. And that is less than or equal to a n plus one times l to the K was on minus one, which you feel we're cool. Um, I believe is this some have the our accomplice important Siri's. And so with this inequality in mind, we can do the same thing we did above rewriting everything in terms of Siri's. So we have a person end. He's equals who? Okay from and plus one for two races King countries listed or equal to K equals impulse one to infinity for a and postman. And again we're writing l, uh, Caymus one which, like in the love problem, can also be simple by Teoh Some preparatory J Valley which goes your to infinity. Then we have a impulse Moved And if you put an impulse, one will see that the first term starts a J and then goes on the wards to Infinity, sir, just because also the J and what this does is pretty much the same thing we had, um, right above where we can now say that because the sequence articles are in close mood is less than a one and converges, We can use this thesis inequality here to basically set of the gym teacher for me, Logan. So what we read, it equals zero. That's insanity. For a and plus mourn times else of A. J wishes a in close moon over one minus. And so you can see it's just the same thing as the human one, which we used above.

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