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JH
Numerade Educator

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Problem 6 Medium Difficulty

Determine whether the series is absolutely convergent or conditionally convergent.

$ \displaystyle \sum_{n = 1}^{\infty} ( - 1)^{n-1} \frac {n}{n^2 + 4} $

Answer

Conditionally convergent

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Video Transcript

for this problem will show that the Siri's is conditionally conversion by showing that this Siri's over here. It converges but does not converge. Absolutely So first, let's go ahead and show that it does conversion. So to do this first we see that this is an alternating Siri's so we can try the alternating Siri's test. No. So to do this, we look at not the whole term here, not the whole way in. We just look at the absolute value there, so we just look at and over and squared, plus four. Then, for the alternating Siri's test, there's three conditions that we must satisfy. The first condition is that being is apartment on negative number. This's always true. You have positive divided by positive condition too. We need that the limited bien as N goes to infinity. A zero what? And this is clearly true. For our example, you could do open house rule here, for example, to see that this limited zero. And for the third condition, I'LL go to the next page for the third condition. We need that the sequence being is decreasing. So what that means is we need to show that the next term in line is not any larger than the previous term. So we can either try to show this by hand or is mentioned in the textbook. We can define this function f by making f event equals being and then we'LL So the difference between F and B is that we could also define F at any real number. So here this is our F. This is the find of a real numbers, whereas BN is on ly defined for the natural numbers. So here, let's look at the derivative of F to show that efforts decreasing. We want the derivative to be negative. So let's use the question well here. Close it rule. So we do the derivative of the top. That's just one times the denominator minus the numerator times the derivative of the denominator, the denominator. He could basically ignore it because it's a square, so it's positive. So it was focused on that numerator and simplify that and we get four minus two ex Claire and we can see that this is negative. If the numerator is negative and you could simplify this to get two is less than X square. So I'm going to the next page there. So with word, this is bien is decreasing if we have and square bigger than two. So there will just take and to be bigger than squared or two, however, since and is a national number. Well, just round up to two. So therefore, by the alternating Siri's test, we we've satisfy all three conditions for being the original series, which all right down here this was on page one. This will converge. No. Now, for the next step, we check whether let's switch color here, whether it's absolutely commercial. So in this case we look at we'LL go into the next page here in a second. But this time we're looking at the absolute value of the original Siri's. Okay, so not the absolute value of the whole Siri's, but the absolute value after you put on the inside. So we take absolute value there on the inside. All that's gonna happen here is that the negative one goes away, and now we have this new series here, and the question is, is whether this thing convergence. So let's go to the next page. Let's try the limit comparison test. So we're doing this test on this new Siri's. So here, If this is my Anne, I should take my B end two b and o ver and square and then that simplifies toe one over in. Then I look at the limit and over bien so limit of if I simplify this and then if you take that limit, you could use low Patel's rule here. If you want, that limit ends up being one. So now we can use the Lim comparison test here. So since bien, this sum is just one over m this diverges. This is the harmonic series. So in other words, you could also do pee test with Pete. Um, P equals one one over into the first powers of P is one. So by the will be comparison test our Siri's that we get after taking absolute value that also diverges. So we'll summarize on the last page here. Therefore, the original Siri's is conditionally conversion and that's our final answer