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Nl? . 3" (F1)" (2n + 1)1Apply the ratio test toPart 1: Evaluate the LimitEvaluateOntl limPart 2: Conclusionn2 The series '(F1)" (2n + 1)!...

Question

Nl? . 3" (F1)" (2n + 1)1Apply the ratio test toPart 1: Evaluate the LimitEvaluateOntl limPart 2: Conclusionn2 The series '(F1)" (2n + 1)!

nl? . 3" (F1)" (2n + 1)1 Apply the ratio test to Part 1: Evaluate the Limit Evaluate Ontl lim Part 2: Conclusion n2 The series '(F1)" (2n + 1)!



Answers

Find the indicated limit. $$\begin{aligned} &\lim _{n \rightarrow \infty} \frac{1+2+\cdots+n}{n^{2}}\\ &\text { HINT: } 1+2+\cdots+n=\frac{n(n+1)}{2} \end{aligned}$$

Let's use the ratio test to see whether or not the Siri's converges. So let's to note this by an now. The ratio test requires that we look at the limit and goes to infinity a flu value a n plus one over an. Now let's go ahead in the numerator and plus one. All right, here will have to in plus three than in the denominator. We just have a n no. All right, so here, let's go in and simplify this. We could cancel and of these threes here these negative threes on one left over. It's absolute value so that'LL just become three and then this will come up to the numerator and then we still have this in the denominator. Select wants the next patient because I'm running out of room. What? This was our limit. Now we can go ahead and rewrite this in the denominator weaken right, this is too, and plus one factorial and then two in plus two, two and plus three. And then we could cancel off the two and plus one pictorial. And as we take that limit, we have three up top and the denominator goes to infinity. So this is equal to zero because this is three over. Infinity equals zero. The women is general. This is less than one. So we conclude the original series on page one. Good. And write that out. Factorial. This one converges by the ratio test, and that's our final answer.

Let's use the ratio test to determine whether the Siri's converges are diversions. Now let me denote this by an then you could go ahead and right out the following for the ratio test. Now, in this case for the numerator, this will be a little more delicate. So we have to go ahead and replace all of the ends in the formula and the expression with n plus one. So here and the numerator will have to end plus one and then and plus one pants Auriol. Now, if you take a look at the denominator, this will turn into the product all the way up to now replace and within plus one. But this term here is just three in plus five. So that means that when the denominator will have five, eh? And we'LL also have three and plus two right before we have three and plus five. And the reason I know three and plus two will appear is because each because the terms air increasing by three each time and these two last two terms differ by three and we'LL see in a moment why that's important. Why, that's an important fact now in the denominator. We just have our usual and here. So just go ahead and put that back in. Now, as usual, when you're taking one fraction and dividing by another, you could just flip that fraction over the blue fraction and then multiply. And you could also see there's no need to write the negative one to the end power because we're taking absolute value and all the remaining terms air positive. So we can actually stop writing naps in value mar Now in plus one factorial, let me replace that with, in fact, for riel times and plus one. Let me justify that by definition and plus one factorial is just a product of the first and plus one numbers. And if you just group the first and together that's a product, they're multiplied. Not not Sorry. Looks like a subtraction, but that this should be multiplication here. And then we see that that first term is just in factorial. So that justifies this over here and then Denominator, I'll keep that as it is. And then now that we flipped the blue fraction upside down and now you can see why was important before till right out this three and plus two term. And the reason for now is because we can cancel this five eight all the way up to three and plus two we'll cancel nicely with the same terms in the denominator. Also, the reason for using this fact over here about the factorial Sze is that I could cancel this in factorial with this other in factorial and we have some more cancellation here. We could take off this tune to the end with this one up here and we're still left with a two on top. So we'll end up with Lim and goes to infinity two still haven't plus one there and on the bottom, all less love left over is three in plus five and at this point you could replace the ends with the X and use low Patel's rule if you want. In either case, you see that this limit is two over three, which is less than one. So we conclude by the ratio test that the Siri's convergence and that's our final answer

Going to problem number 67. Here we heard divide direct limit and then student Fandy who are plus two plus three up to plus a year end by you quit that is limiting and pence to infinity. It is a new and press one by a group governor but he and square so limit then danced in pretty in and apparently calmer plus one by yet they were Bankole ordered by in for so in Esquire and Scarlett get cancer They're not play limit It is leaving my intention frankly, one by doing do one plus one by yet if you play here on my toe and one plus learned very invented when by doing Go on. Plus there are logistical look by and by Thank you

Correct. We want to find the and partial some and evaluate its limit to determine whether or not the series converges. The given series is some N equals 12 and many of one over N squared plus three plus two, which we've already written as the sum from n equals one to infinity of one over N plus one minus one over n plus two. To find the answer, partial sum, we need to determine how this series changes from terms terms from N equals one to n equals two and so on. So the and partial sum is SN equals something Michael in the end one over N plus one minus one over N plus two. If we look at this term, the term the very first term for equals +11 over N plus one would be saved. But in each alternating terms for equals one of one over N plus two and equals to one over n plus one terms cancel. So the and partial sum can be expressed as one half minus one over N plus two. So we can now take this limit as N approaches infinity. The women as N approaches infinity. UsN is limit, an approach infinity of one half minus one over to rather one over N plus two. This is one half minus zero equals one half. So since that limit converges, the series must converge to value 1/2.


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