4Use any method to determine whether the series converges: 41 )>Jutioi...

Use any method to determine whether the series converges. $$ \sum_{k=0}^{\infty} \frac{(k+4) !}{4 ! k ! 4^{k}} $$

So in this problem we are asked to use any method to determine whether this series converges and we're giving us some from one to infinity of Caples Four factorial over for factorial K. Factorial times for the K. So let's use the ratio test here since we have factorial. So that means we take the limit as K approaches infinity of a sub K plus one over a sub K. The sequence here being equal to a sub K. Okay, So that means we have the limit as K approaches infinity of and we just replace K with K plus one and it's okay to get a sub K plus one. So we have K plus one plus four factorial over. For factorial times K plus one factorial Times 4 to the K Plus one. And this is all over. Case of K, which is just uh Okay. Plus four factorial over. For factorial K factorial for the K. Okay, so now let's multiply by the reciprocal and we get that this is equal to the limit as K approaches infinity of Cape was five factorial over for factorial, K plus one factorial For the K-plus one times four Factorial K Factorial for to the K over K plus four factorial. So now recall what a factorial means. If we have K factorial, if we let's say we have Cape was four factorial as in our case That is equal to K-plus four times K plus three Times Everything. All the way down to one each term. And that means we have K plus five factorial. That's equal to K plus five times K plus four Times keep us three all the way down to one. So as you can see, Cape was five factorial is just equal to K-plus four factorial as these terms here Times K-plus five. So let's replace with that. And our equation. So we have the limitless K purchase infinity of K-plus five Times K-plus four factorial over four factorial. And we can do the same with a K plus one factorial. It'll just be equal to K plus one times K. Factorial Times for to the K- Plus one. Yeah. And this is all multiplied by four factorial. K factorial for the K over. Okay. Plus four factorial. So now we can do some canceling, you can take out the K plus four factorial, you can take out the K factorial. We can also take out these four factorial and we're left with the limit as K approaches infinity of Okay. Plus five Times 4 to the K Over K-plus one Times 4 to the K. Plus one. Now we know that if we have exponents like that that we can rewrite them. So on the top if we leave it the same from the bottom, we change for the K plus 1 to 4 to the K times for the one. And now we can cancel these four to the case and we're left with distributing the four, We're left with K-plus five over four K Plus four. And so now we can take this limit and we see that these K's Are the highest powers and their coefficients are one and 4. And so this limit is equal to 1/4, Which is less than one. So in accordance with the ratio test, that means that this series converges, so the sum from one to infinity of K plus four factorial over four factorial K factorial times for the K converges by the ratio test. And that is all that we have to do for this problem.

This is on the forms and over for our and from one to infinity. So let me try to test the absolutely convergence. So we have considered it's so much in absolute Saiful and other for well and from one to infinity And we were the same as the absolute aside for and everybody phone bell. And now I used the come person that's yeah and we noticed that the afternoon decide for and it would be smaller than one. And therefore we have the absolute Asai for an over farmer and will be smaller than 1/4 about and on the owner hand. We know that the submission of the one fall well and is convergent by no Geico romantic Siri's and therefore bind a comparison test Yeah, we conclude that disagrees is convergent by this one And because in this converted in that and we imply that this tree on the top there will be absolutely convergent

In discussion we need to use any appropriate test to determine whether the series one upon three plus two upon 4 plus three upon 5 plus four Upon 6 plus so on converges. So let's see how to fall discussion, Consider the first time. Once upon three if this institute okay on the place of one So the denominator three will be close to Okay plus two hands one upon three can readiness K upon K plus two. Similarly, If we consider second term to upon four and if you take K is equal to two. So K-plus two becomes 4 hands two x 4 can also be tennis k upon K plus two. So when we continue in this manner so we can write the given cities as submission. Okay is he calls to one to infinity K upon K plus two. And this series is a cities of nonjudicial firms. Therefore on the basis of the cities we can write A N plus one it recalls two N plus one upon N plus three and A N equals two and upon N plus two. Now we have the values of a interest one and A. And so we can now apply the ratio test to determine whether the series converges. So let's discuss the ratio test samples submission. Okay? It because to one to infinity E K G is a. Mhm. Cities of Yeah positive terms such a bed limit. Mhm and tends to infinity A and plus one upon A. And the vehicles to L. Yeah. So mm if L is less than one so we can conclude the cities is convergent. The second possibility is if L is greater than one or if L is equals to infinity. The mhm series is divergent. Yeah. And the third possibility is if Ellis recalls to one the ratio test his in Yeah, conclusive. It means we cannot decide whether the series converges or diverges. Yeah. So this was a ratio test. Now let's find the ratio of A and plus one upon a and and this will be close to And plus one upon and plus three upon and upon and plus two. So this will be calls to And plus one upon and Plus three into and plus two upon. And now let's take out and as a common from the each term of the numerator so we can write and esquire into one plus one upon end into one plus two upon end upon let's take out any square as a common from the denominator. So this will be an esquire into one plus three upon and the ministry will be cancelled by the any Squire hands. Finally we get the ratio one plus one upon and into one plus two upon and upon 1-plus 3 upon and no by the application of ratio test we can right limit and tends to infinity A and press one upon ian because to limit and tends to infinity one plus one upon and into one plus two upon and upon Yeah one plus three upon end. Now substitute infinity on the place of end. So we get one plus one upon infinity into one plus two upon infinity upon one plus three upon infinity. So this will be calls to one Since the limit is equals to one. Therefore, the ratio Yeah, test is in conclusive, which means we cannot decide whether the given series converges or diverges. So this is a final answer for this problem. I hope you understand the solution. Thank you.

Hello. So here are given series is the series where we have K. Going from one to infinity of the greatest has won over K to the 3/4 power. Now then by while divergence of the harmonic series we have that are given series here is going to be divergent since we have that one over the cube root of K. two. The 4th plus four is going to be greater than or equal to one over K to the three force. And this is for all K greater than or equal to one. So therefore our given series um won over the cube root of K two. The fourth plus four is going to diverge.