(12 points) Determine whether the series listed below are convergent Or divergent: Give the sums of the convergent series:2 2 7+4n+3 E"() 2 Inn - In(l + 2n)....

$2-28$ Determine whether the series is absolutely convergent,
conditionally convergent, or divergent.
$$\sum_{n=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2 n)}{n !}$$

So If we let s event equal negative one to the end -1 Times and over and squared plus four, then this problem asks about the series, the sum of Osama bin from N equals one to infinity and were asked about its convergence absolute or condition. So first let's look at the suburban which equals the absolute value of ace event, which simply equals And over and square plus four. We need to look at the sum from N equals one, two in friendly of this and we need to ask if this is convergent or divergent. If it converges, we know that The sum of a seven converges absolutely. So the sum of an over And squared plus four does it converge and diverge. The answer is this series diverges by the limit comparison test with the standard harmonic series one over end, which we know diverges. We can see that if we take B seven and we divide it by one over N, then what we get is n squared over and squared plus four. This approach is one as N approaches infinity, and therefore by the limit comparison test Some a B seven diverges. It behaves the same as one over, so That means that the sum of a seven does not converge. Absolutely. So the question is does it converge at all? And the answer is yes. It does. It converges by the alternating serious tests just growing up and taking a look at it. The non alternating part approaches zero as an approaches infinity. So the some of that series converges by the alternating serious test, which ultimately means combined with the fact that it does not converge absolutely. It converges conditionally.

Look at the series where you are determining the convergence or divergence of this. And if we're looking at convergence, it's either absolute or conditional conversion that we need to specify. And so let's go ahead and start by take the ratio test. So we have 10 to the endless one over and plus two. That's four to the two N plus three power times are divide that by since at the end over and this one terms 2 1 Plus one. Okay. So that going from here got the limit. Ask protests infinity tend to the N plus one over N plus two 4, 3 times four to the two N plus one, seven plus one intensity. And sorry. Okay. So then from here, so we see that on this is tenure. This gives us four squared. This goes to one. All right. So that means that we have 10 Over four square age 16 and that is less than one. So therefore series that we have absolutely coverage bi racial justice

Our objective is to see if this is absolutely convergent, conditionally convergent or diverges. And so let's go ahead and apply the ratio. Test to check that out. So we take the limit as N approaches infinity, you can ignore the negative one to the end power here because we're in the value To to the end plus one and plus one factorial multiplied by. So we'll have five times eight times 11 and then all the way to three and plus two times three plus five, correct? So now normally we divide by what we started with but since we're dividing by what we started with here we can multiply by it's reciprocal. So five times 8 times 11 Times all the way to three n plus two. And that's all divided by 2 to the end times that factorial. Right? So canceling out here, we've got two on top Time Zone Plus one cancel with it to the end. And in factorial that we've got all the way from The multiplication for up to three and Plus two cancels out with what we have there. And so then we see that what we have here. His so we've got two times and plus one. So it's like two in The bottom here is three in take the limit as N approaches infinity here And that gives us 2/3 since it's supposed to the and power I mean to the end of the power of one. And so that's 2/3 which is less than one. And so therefore this series is absolutely convergent because what we took the opposite value with the ratio test, we were also taking a look at the whether it's absolutely convergent if it does conversion, so therefore we have that.

Check and see if this series is absolutely convergent, conditionally convergent. That Richard. So let's go ahead and apply the root tests. It has to the end power there take the limit as N approaches infinity of the answer right? Hoof squared plus one Over two and squared plus one to the and of power for each of those. And so he end up having the limit as N approaches infinity Event Square Plus one Over two in square plus one. So because they have the same power top to bottom here, we just take the coefficients. This is one over to here, So it equals 1/2 because there was only one half, which is definitely less than one. So therefore we would say that our series is absolutely convergent.