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JH
Numerade Educator

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Problem 12 Medium Difficulty

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {7^{n + 1}}{10^n} $

Answer

The sequence converges to $\frac{49}{3} \approx 16.33$

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Video Transcript

let's find at least ten partial sums of the Siri's. So that's for example, we could do us one as to all the way of us ten. So here this is our and Anne is seven to the n plus one over ten to the end and then es end. By definition, you start from your starting point of the song, which in this case, is won Scott. I equals one and then we go up to end of a So, for example, as one is a one es tu is a one plus a two and so on. So here I'll find the ten partial sums by just graphic them in the copy later. So here you could see my end's term. A end is seven and plus one over intensity and just how we want it. And then down here, you see the same man rotation. So and and I'm letting the Sigma go all the way up to ten, you could see that little and equals ten up here. That's the upper bound. So here we go ahead and we see the graph. So the first partial some is four point nine second partial, some a point three three and all the way up into the tenth. Partial some, Which is about so these are all approximations to three decimals. Well, there's at least this last one is so fifteen point eight seven two. So here you can pause the screen and write those down. That's the first ten partial sums. The first quarter me tells you how many things you're adding. The second term is the sum for that value. And so that takes care of the first part of this question that wanted to see the approximations or the values of the first ten partial sums. So that takes care of the first ends. Now for the second part, bull graph the end. That's a sequence of terms, and then the sequence of partial sums. That's the SN! Let's crackles on the same screen. So coming back to our graphing calculator, let me remove this label and then let me go ahead and purple. So let me temporarily remove the red graph so purple. This is this sequence of the AI ends, So a one is about five. That's four point I'd aides who is about three plane for three, and so on. All the way up to eight ten about point to it looks like they're getting close to zero. And then now it's Graff, both of them on the same screen. So here the red values were getting bigger, so I will have to zoom out a little bit. So there. This is a little tricky because although the sequence looks like it's going to zero, the sun does look like it. It is getting bigger. So the question is whether this red grab the sequence of sons will continue to increase or will level off. And here we could take advantage of the fact that run a graphic cockle Idris delicious increased the number of terms, for example, that of doing ten terms. We can go toe a hundred terms and we save a lot of time. We don't have to do this by hand, and now we can see it looks like the purple graph. Let me remove the labels there. The purple graph. The sequence of the AI ends is getting closer and closer to zero, and it looks like the partial sums air leveling off to some number a little bigger than sixteen less than seventeen. So here based on the graph. It looks like it converges. So that answers this question here. Now. If it is conversion, let's go ahead and find that some. So we want the sum of the Siri's and equals one to infinity seven and plus one over ten CNN. Now this is geometric and away to sea. This is to rewrite eight of the end. You just pull out of seven and then so that's a ten down here, tend to the end. And then this is seven times seven over ten to the end power. So now we see that this is geometric. So no, no dividing there. But the some stars from one to infinity. So this is geometric. We see our equal seven over Zen. And since this value of our satisfies the inequality that it's an absolute value less than one that's hoses, that it does actually converge. So we were correct based on the graph and then since his geometric, we have a formula for the sum. So recall you to the first term over one minus R So I'LL go on to the next page to simplify this. So the first term when you plug in n equals one. So we're getting seven times seven over ten to the one that's the first term and then one minus R. So that's one minus seven over Sen. So that's forty nine over ten, and then we have three over ten on the denominator. So go ahead and simplify that. And we're getting forty nine over three, which agrees with the graph that's a little bit bigger than sixteen, but less than seventeen, and this is our final answer.