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Determine whether the series is convergent or divergent; 2 n = 9 - e-n convergent divergentIf it is convergent; find its sum. (If the quantity diverges enter DIVERG...

Question

Determine whether the series is convergent or divergent; 2 n = 9 - e-n convergent divergentIf it is convergent; find its sum. (If the quantity diverges enter DIVERGES.)

Determine whether the series is convergent or divergent; 2 n = 9 - e-n convergent divergent If it is convergent; find its sum. (If the quantity diverges enter DIVERGES.)



Answers

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {e^n}{n^2} $

Back to New Murad. My name is Kevin Chirac. Let's take a look at the infinite Siri. This which is? Let's see the summation from any cools one to infinity and on the inside will have 10 to the end. Power over negative 9 to 10 of the net. And over Negative nine to the end, minus one power. Now, we could do a little bit of manipulation here to make this look like what we are used to seeing in a geometric series. Let me show you what that might look like. We could say and equals. One could still be on the bottom for right now we're going, Teoh. Want to kind of arrange this into being a common exponential expression. So how could we do that? We could write this as 10 times 10 to the end, minus one divided by negative nine to the end, minus one. And then we could write this more simply as an equals one to infinity of 10 times 19th. That's include that negative sign Negative 10 nights to the N minus one. Now notice that one thing we could choose to dio is we could change the index. We could have this entire thing start at zero and infinity, make this 10 and then make this negative. 10 over nine to the let's see, if he had plugged in one before, we would have got zero. So this would just be to the end now, right? Well, this fits in very nicely with how we want to calculate. Sums are partial summation would normally fit into this structure. One minus are where we have our and we have our A. But notice that right now are our value is greater than one is absolute Value is greater than one. That means that this is actually gonna continue Teoh jump around much more sporadically as we add, more and more terms are partial. Summation is not going to be a serious that converges. So this is actually going to be a serious that diverges. In other words, this equation does not hold. And we would say simply that this serious diverges. I hope that video helped if it did take a moment and click that heart at the bottom of

Two New Murad. My name is Kevin Chirac. Let's look at the infinite Siri's from n equals one to infinity, defined by one plus two to the end over three of the end. And we're curious whether or not this serious convergence. Well, one thing we could look at is we could do the 10th term tests. We could run a limit test, honest, and I encourage you to do so. So we do. And as an approaches infinity of this inside term here But before even getting too far into this, you should see that this this number here is gonna follow a similar pattern to the problem that we just did before. So let me show you what I mean. This can be rewritten and regrouped so we can have a limit as n goes to infinity. And we could write this separately, we could have won over three to the end, plus 2/3 to the end. Good. Well, what's gonna happen? This is actually going to drop off to zero, as is this one. So, in fact, as end goes off to infinity, this is going to be zero. Okay, so it means not that we necessarily converge, but that we don't diverge well, let's continue with a couple other ways that we can manipulate this expression. Then we'll break it down. Perhaps we could write this as the summation from n equals one to infinity of 1/3 to the end, plus to the end over three of the end. We'll keep that This separated would be the advantage. While the advantage would be that I could break the summation into to remember that this summation notation is nothing more than a compressed version of addition. And there's no reason why I can't separate addition into two different sets of addition. Whether it's an infinite amount of addition or not, I have the ability to perform certain parts of the addition first, at no consequence. Well, you might notice that I'm writing this second term toe look very much like a geometric series. So let me take a moment and see if I can calculate with that might be, remember the a term that is an hour our former for the summation. So it's a over one minus are are a term is going to be the first term. So if I put in n equals to one. I would get 2/3 and one minus R will. My art is gonna be that 2/3. We do a little bit simplification here, multiply through the top and bottom by three. And we would get to over one or two. That's gonna be just that second part. Let's also now do the first part. Well, it's gonna follow a very similar process. A the eighth term, the very first term is going to be 1/3 and then we're gonna have one minus R where r is that common ratio and we can see here that that's going to be 1/3. So if we multiple life through by three, then we're going to have one divided by we're gonna have yes, one divided by two. So this entire thing is going to come out to 1/2. Well, now we just simply need to add the green and the red. So if we add the green and the red perhaps will use, let's use blue, and that's just going to be 2.5 would be our final summation for the this problem. So I hope that was able to help out if it was, take a moment and click that heart at the bottom of the video to let me know when to help me reach out to more stuff Dunes just like you.

Welcome back to New Murad. My name is Kevin Chirac. Let's take a look at the infinite Siri's that is defined from an equals one to infinity by one plus three to the end, over to the end one plus the read to the end over to then pence Getting away from here now what can we tell Week are looking to figure out whether or not this Siri's converges? There's a couple of different tests that we could run, one that looks like it might appear, since this doesn't appear to be geometric would be the end of term, test our limit of our inside argument and to figure out whether or not that inside argument is eventually going to become insignificant. What we contest that by doing the limit as n goes to infinity of one plus three to the end over two to the end to figure out if our quote unquote for ever addition in this series is going to be nothing, a k we've converged possibly, or it's something. In which case we've definitely not converged, but we could rewrite this out just a little bit so we could say the limit as n goes to infinity of 1/2 or one over to the end. Excuse me. Plus the read of the end over to to the end. And what we're going to see is that this part here is going to drop way to see room, which is great. But this part here is going to be increasing all the way up until infinity. So our limit here is going to be infinity cause I have something. It's greater than one that is to a positive power. Well, that means that there is no l. There is no equality here that this Siri's is equal to you could write it as it's not equal toe any L.

Let's determine whether or not the Siri's conversions. So here, If we notice that four over and over either then you can rewrite this. And if we take the limit, this will go too Infinity as n goes to infinity since four over e is bigger than one room is about two point seven. So if you take a number larger than one and keep multiplying it by itself and you go to the limit, this will go to infinity So we could see that because of this Since this is bigger than or equal to for the end over either then and we just showed that this term goes to infinity. Since this is larger, this must also go to infinity in the limit. So we have lim and goes to infinity to end put Teo and plus four the end over either of them. And this will equal infinity and we use the test for diversions. So I hear you're given a Siri's and your terms or an so if the limit of am does not exist or if the limit does exist. But it's not equal to zero, which is our case Here we have infinity This is not equal to zero, then the Siri's will diverge. So our case, we took the limit. We're using the test for diversions. We took the limit of our way in here. Delimit exist, but it's not equal to zero. Therefore, by this test for diversions, our Siri's will diverge. So I should put diverges, and that's a final answer.


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