2024 AP Calculus BC Exam FRQ #6

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Free Response Question 6 from the 2024 AP Calculus BC exam.
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The series question!
Given a Maclaurin series we have to determine convergence of a series; use alternating series error bound (show that an alternating series converges); find the general term of f', the derivative of f; use the ratio test to find radius of convergence twice! (power series and its derivative or integral has the same radius of convergence)

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@CalculusConnoisseur338 2 ай бұрын

As someone who skipped Calculus AB and started Calculus BC from Precalculus, i would like to say that your videos are the reason why I have an A in my class and a 5 on this AP test. I really appreciated the short videos you posted a few days before the AP exam as there were definitely questions on the test where i could apply your insight.

@legoatjagoat4691 2 ай бұрын

Skipping ab is normal no? Ab is a part of bc

@Hyzpen 2 ай бұрын

@@legoatjagoat4691definitely not normal but sure

@nak_attak 2 ай бұрын

from helping me with AB last year to helping me with BC this year, turk, you were a major catalyst in me getting a 5 on AB last year and undoubtedly a factor in getting a 5 on BC this year. thank you forever!

@ashvinjaishankar3813 2 ай бұрын

I think for 6c, you can say that the radius of convergence for f and f’ are the same but like you I did the ratio test to cover my bases.

@turksvids 2 ай бұрын

that's definitely the way they meant for you to do it! i totally forgot about that idea (I think I like the ratio test a little too much...) and thought it strange that they were making us use the ratio test on consecutive series. I feel silly having forgotten, to be honest. Feels like using the ratio test on a geometric series and then testing the end points only to notice geometric.

@joekarbowniczek6938 2 ай бұрын

This is what I did as well. Glad I didn't make up that idea 🙂

@ashvinjaishankar3813 2 ай бұрын

I think error bound is the one exception to not simplifying arithmetically or at least that’s what I’ve been noticing.

@7rook768 2 ай бұрын

i’m so annoyed i did the interval of convergence for both C and D. i really hope i get some kind of points for that because i got it correct both times just with the interval and not the radius

@turksvids 2 ай бұрын

I think you will get quite a few points. The process of finding the interval does involve finding the radius at some point. You do have to explicitly call out "the is the radius" to get the final radius point, though.

@aaravshah3843 2 ай бұрын

For part c, couldn’t you just say the radius is 6 without doing the ratio test, since the radius of convergence for the series for a derivative/integral is the same as the original?

@elicoe9964 2 ай бұрын

Yes, I would hope the College Board would say that's okay. The only thing that will really change are the endpoints, when taking the derivative or integral. So I figured they were testing knowledge on the interval of convergence, and instead of staring at the clock for another 30 minutes, I calculated the radius of convergence and interval of convergence.

@ericchavez8465 2 ай бұрын

For part (c) there is a theorem that states the radius of convergence for f and f' are the same, but their interval of convergence could be different. Since the radius of convergence for f is 6 (given) then for f' it is also 6. I wonder if stating that would be sufficient for part (c) instead of using the ratio test.

@turksvids 2 ай бұрын

yeah, that's definitely the "other thing" i couldn't think of while using the ratio test the second time! would have saved quite a bit of time to use that fact!

@Dollaboi792 2 ай бұрын

Would I get points off if I just did the ratio test by adding plus one and then dividing by the regular or would I have to write the full absolute value and limit form out

@rinnoX1 2 ай бұрын

gotta find the limit of that

@elicoe9964 2 ай бұрын

You will lose points if you do not show the limit as n goes to infinity and also include absolute values. However, you will still receive credit for arriving at the correct answer. They want to ensure that you are prepared for Calculus 3, so the short answer is yes, you need to demonstrate that you know how to perform the ratio test correctly.

@Zzzyszzes 2 ай бұрын

I used the integral test for A and I think I messed something up when I did the second question because I used the 4th term but I think I saw (-1/2) as (-1) and that gave me the wrong answer. For C I got |x|/6

@turksvids 2 ай бұрын

for radius of convergence you always want to work until it looks like |x-c|

@vustormz8461 2 ай бұрын

Could I have used direct comparison test? I literally just simplified it to 1/n and said it diverged by comparison to divergent harmonic series

@turksvids 2 ай бұрын

n+1 > 1 so (n+1)/n > 1/n and direct comparison does work, i'm pretty sure. (I am not a fan of direct comparison, to be honest, because I don't like showing one series is bigger/smaller than another if it's not straight forwardly obvious)

@vustormz8461 19 күн бұрын

What if I didn't make the statement outright but just knew one was bigger than the other and just did the test

@nak_attak 2 ай бұрын

a) a few of my classmates and i did dct with u/u^2 and got diverges also b) i completely forgot to list the conditions for alt. error bound but did the question right c) i somehow got 1?? i don't even know d) i somehow got 1 again??? 😬

@andrewzhang3500 2 ай бұрын

i did D.C.T for a as well, i think both works.