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Use ratio lest t0 detenine if the following series converges diverges: 3"+2n-5 MarksUse root test l0 determine if the following series converges or diverges: 2...

Question

Use ratio lest t0 detenine if the following series converges diverges: 3"+2n-5 MarksUse root test l0 determine if the following series converges or diverges: 2"n35 MarksLet f (x) =then find the first three terms of the Giled series generated byMarks

Use ratio lest t0 detenine if the following series converges diverges: 3"+2n- 5 Marks Use root test l0 determine if the following series converges or diverges: 2"n3 5 Marks Let f (x) = then find the first three terms of the Giled series generated by Marks



Answers

Use the Ratio Test to determine whether each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{n 5^{n}}{(2 n+3) \ln (n+1)}$$

Let's use the ratio test to see whether the Siri's is conversion or vibration. So let's go ahead and do knowthis by n then the ratios has wants you to look at the limit as n goes to infinity, and then you take the absolute value A and plus one over a end. So notice that the plus one is because the one is being added to that and not a M. So now let's go ahead and see if we could simplify this and plus one in the numerator That's just and plus one over five and plus one and then in the denominator years on end so and over five in Now let me bring this down over here so I don't need absolute value here because all the numbers and the fractions are all positive. So here we have n plus one over n. And then let me write this as I have five end and then I'LL rewrite this term here using my exploded properties I can rewrite This is five times five to the end and then I'LL cancel those five ends and that here. So I have a five on the denominator and then as I take the limiters and goes to infinity this fraction, if you want, you could use a little bit, tells Rule here. If I take the derivative of the N plus one, that's just one. If I take the derivative of end with respect to end issues one and then we still have this one over five, so we get an answer of one over five. This is less than one. So we conclude that the Siri's the one that we started with convergence by the ratio test, and that's our final answer.

In this question. We re contraband ratio test. Okay, Okay. Which considered a limit after I am bliss One How What? I am Invest your infinity Supposed this limited could, um And if the area smaller than one and then we got the streets will be convergent greater than one on. Then we get the serious. It will be divergent in co 21 and then it will be in conclusive in this question here were given the submission off the job in in about five. Get someone to infinity the first time we need to compute the limit of the I am just one hour AM Ingestion infinity! We can continue our limit. E am this one again to Cuba And this one over m just one power five and then divided by i n. We need to multiply by reciprocal. We end about five hours to power and we can consider tuba. And with this one to bring the two outside the limit, you can limit off the and power five over and plus one hour, five and gusty infinity. You noticed that they have the same power five Therefore, this limited. Go to the leading coefficient here with two times going. Number one in could teach you. And two, he is greater than one. They're far from the Park Chu. We conclude that these trees here it will be divergent.

Question would record about the ratio danced which consider the limit. After I am this one off I am and goes to infinity. Suppose this limited. Go to L. They will have the m here smaller than one. The industries will be convergent greater than one. The serious could be divergent Equal to one desire is gonna be inconclusive in this question. Here was given the serious number farm to end for tire on the five power and from one to infinity Let us stand the first time We need to find a limit off the I am this one off a n and just to infinity. Then we get a good Jew The limit off Now I am just one going to two times endless one factorial divided by fiber And this one divided by i n We need to multiply by reciprocal five out and I want you and factorial and goes to infinity. Notice that we will cancel the fiber and with this power joined the two and factorial with this for tore your hair. So we have left with a limit off the and plus 1/5 and goes to infinity and its limit eager to infinity is bigger than one. Therefore, from the second part here, we can conclude this seriously will be divergent.

In this question, Will you call a band racial test which considered a limit after I am just one over the i m and goes to infinity? Suppose this limited coach email. And if the air here smaller than one then we got the stories will be convergent greater than one. Then the streets would be divergent because you want industry is gonna be inconclusive in this question. Here we are, given the threes on the form and times five an hour and factorial standing themselves to infinity. And here I noticed that the first time we need to find a limit off that I am this one. How? But I am and goes to infinity. They're going to limit off. I am just one gonna I am just one then, fiber. And this one hour, the m just one trial dividing by n We need to move by by reciprocal we n factorial off anthems, firepower and angers to infinity. Notice that we can console, uh, this and factory with this factorial The endless one would end this one fiber and with five with Bauer here. So we have never heard only the limit off the five hour and and goes to infinity on this limit here, equal to the cell smaller than one. They're far from the first part. We can conclude this is we will be convergent.


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