Problem 43 Hard Difficulty

(a) Use (4) to show that if $ s_n $ is the $ n $th partial sum of the harmonic series, then
$ s_n \le 1 + \ln n $
(b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22.

Answer

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Video Transcript

here s said is defined is it's a sequence where the term is define is the ends partial some of the harmonic series for part A Let's go ahead and show the following inequality That s n is no larger than one plus natural Aga Ben Okay, so first, let's go ahead and start a graph here. So first I'd liketo plot some rectangles here, corresponding to the terms of the Siri's one over it. So the first term is just one. So represent that with a rectangle with one in Hye won, all my red tingles will have lengthened with one and then the height will be one of ren. So the second right angle that height will be one half and someone and let's also draw the curve one over X. Where did that come from? Well, it's just a continuous version of an replacing and with X and this more or less looks something like this, you know, it's always above the rectangles, but it will touch at the right corner. So if we look at just looking at the picture here, we can see one over end, which is the area of the end rectangle. So one of her end is a rectangle over the interval and minus one to end on the real line. And we know the height is one over end, and we know that the curve will touch it at the right corner from above. So the area of the rectangles, just a n that's in blue but then in red is the right hand side. That's all the area under the curve over the interval. So that's why the red area is larger than the blue. So that implies this inequality here, if you your ex interpreting these terms, Azarias, that's what the trick is here. Okay, so let's and also here when I'm using this inequality, I'm using it for an at least two because otherwise we don't want to deal with and equals one, because this is not a conversion in earlier. This is improper, integral diverges because you can see here the current goes up forever. You have to write this. Integral is the limit, and your answer will be infinity. So that's why I'm only doing this inequality for and at least two. So let's go ahead and use this in equality for all of these ends and then add them all up. So I'm not writing the whole harmonic series. The partial summer I'm excluding the first her because again and was at least two. And now using that in the quality that we have and minus one times more were again thie inequality that we had was of the form. So we're just using that for each of these corresponding terms and so on. However, if you look at the right hand side, you notice these all have the same Interbrand and we could use a property of integration of intervals that if you add the animal from A to B and from B to C, that's just the animal from A to C. So if we use this property multiple times on the right hand side here, we can rewrite this right hand side is in a girl from one toe end and then DX over X, which becomes natural lot of end. And then now, at once of both sides, so by both sides, I mean the left hand side here, lefty inside and then right inside we get this is less than or equal to, and then one plus natural log event and this finishes party because the left hand side here after adding one, this is now equal to SN. So let's go on to the next page for a party. So we although the Siri's that verges, we'd like to show that it diverges very slowly. So what that means is that you have to add a lot of terms, and even after doing so, you're some won't even be that large. So here will show to an apology's one that if you had a million of these terms, the millions partial some that's a one with six zeros there, one over I or s sub one million. If you want to write that, we want to show that's less than fifty. And then we'Ll also show that if you did a billion terms, it actually is not much larger. In this case, it will be less than twenty two, actually. So these confirm that even though this is infinite sum, it just happens to grow very slowly. Okay, not for part one. Well, we'LL go ahead and use the inequality. So remember before we showed up for a sn one plus natural log cabin, so go ahead and just plug in one million friend and this will be less than or equal to. And then we'LL have just plugging in the same end on the right hand side. Here, let me write. That now is tend to the six and then this is one plus six. Ellen ten and a calculus gives about fourteen point eight, which is less than fifteen. So that confirms our the first art here. Part B. Similarly, using this again plugging a billion for friend And here I'LL just write in the scientific notation That's tense of the Ninth Nine National lock of ten. Now we'LL go to the calculator. This is about twenty one point seven, which is indeed less than the desire twenty two, and that confirms the second inequality.