Power series ultimate study guide

[@blackpenredpen]

Q1, Power Series of x/(1-4x) at a=0, 0:47 Q2, Power Series of x^4/(9+x^2) at a=0, 6:00 Q3, Power Series of (1+2x)/(1-x) at a=0, 10:58 Q4, Power Series of 1/(x^2-5x-6) at a=0, 17:45 Q5, Power Series of 1/(1-x)^2 by partial fractions at a=0, 27:17 Q6, Power Series of ln(1+x) at a=0, 33:17 Q7, Power Series of tan^-1(x) at a=0, 45:56 Q8, Power Series of 1/(1-x) at a=3, 56:02 Q9, Power Series of 1/x^2 at a=-2, 1:04:10 Q10, Power Series of 1/(x^2+6x+10) at a=-3, 1:22:26 Q11, Power Series of e^x at a=0, 1:25:20 Q12, Power Series of sin(x) at a=0, 1:32:50 Q13, Power Series of cos(x) at a=0, 1:43:10 Q14, Power Series of e^(3x) at a=2, 1:46:35 Q15, Power Series of sin(x) at a=pi/2, 1:52:42 Q16, Power Series of sin(x) at a=-pi, 1:59:02 Q17, Power Series of sin^2(x) at a=0, 2:04:07 Q18, Power Series of cos(x) at a=pi/4, 2:09:54 Q19, Power Series of sinh(x) at a=0, 2:20:43 Q20, Power Series of cosh(x) at a=0, 2:27:58 Q21, Power Series of tanh^-1(x) at a=0, 2:31:46 Q22, Power Series of ln(x) at a=2, 2:41:50 Q23, Power Series of 2x^3-5x^2+1 at a=1, 2:48:45 Q24, Power Series of (1+x)^r, i.e. the binomial series, at a=0, 2:56:33 Q25, Power Series of sqrt(4+x) at a=0, 3:12:29 Q26, Power Series of sin^-1(x) at a=0, 3:17:34 Q26.2, Power Series of x^0.2 at a=26, 3:31:50 End Tejava black tea & 2019 Long Beach Marathon Medal 3:34:50 The File: 936933f9-1455-44ce-b414-4d0b35a6c090.filesusr.com/ugd/287ba5_9ffcecc7c15d4605a07b901a19062a44.pdf T-shirts: teespring.com/stores/blackpenredpen Patreon: www.patreon.com/blackpenredpen You can also support me by simply watching & sharing my videos. I greatly appreciate your support. Thank you! Click here to subscribe kzbin.info 100/(1-x), math for fun

[@IndieSamurai101]

I really love your dedication!

[@orvinal2883]

It takes mad dedication to make a video that you no people will only watch 2 minutes of

[@ma-thdimension3474]

🔥

[@anuragguptamr.i.i.t.2329]

Please make a video on Maclaurian power series expansion of e^sinx upto 6 terms at least.

[@antiretro1094]

Welllot of update 7I ip can't I just i I ii. I i7.30pm If my pi and ion iWell I Well baterija I understand i8iiiip. I. If I onon on on i8. pick up. pick up II II and 98 98 to itself iii I iii oil in the wise afternoon wise wise ip0.30pm your I in the own i, of the but oflove and all of. was p was You You the ius in not not. the in 7iand ithat ofget .. 0, you iipcan ice ice 8 in the make a lot of

[@mingbeckwith2232]

I just finished my Calc 2 final and I could not have passed without your help. Thank you so much!

[@blackpenredpen]

I am glad to hear!! Cute dog btw.

[@mingbeckwith2232]

@@blackpenredpen Thank you! She is my best friend! Along with 1/(1-x) of course 😁

[@MathManMcGreal]

Thanks for these so much! Seriously. As as first year Calc BC teacher, these have been SO HELPFUL for my students. They are currently on problems #40-50 in your derivatives packet and we will definitely work through your integrals next. YOU'RE. THE. MAN.

[@blackpenredpen]

You're very welcome and I am very happy to hear that my materials are helpful! Thank you.

[@elif9422]

I'll understand this in about 3 years, and when I do, I will watch this video in its entirety and appreciate it as much as it deserves.

[@subhrajitroy1477]

why in 3 years?

[@elif9422]

@@subhrajitroy1477 that's when I'll be able to take the class

[@subhrajitroy1477]

@@elif9422 ohh...what's ur age now? Where are u from?

[@elif9422]

@@subhrajitroy1477 I'm 16 in the US

[@subhrajitroy1477]

@@elif9422 I am 16 too, in India...but we will take the class next year lol...Class 12 Science and Mathematics curriculum in India is interesting af...😍😍

[@mercenarioo88]

I am from Latin America, and I am 1 year away from graduating, the truth is that the education here is not good, and many things that you explained I did not know, thank God I know your channel :)

[@AntimatterBeam8954]

Your channel keeps my pretty much lifelong mathematically oriented mind active. I can only give £5 as I'm multiply disabled and on government benefits. I hope you are well!

[@blackpenredpen]

Thank you very much!

[@AntimatterBeam8954]

@@blackpenredpen you are very welcome!

[@patrickalava9650]

This is brilliant. In the first five minutes you manage to make radius and interval of convergence immediately accessible and relatable. I am a math teacher and your videos give me the chance to observe your lessons and pick up tips.

[@patrickalava9650]

And also: I just (duh!) figured out why your channel is called "blackpenredpen" and I'm going to use that penholding method now! I've always had to put down one color and pick up another when highlighting steps. Awesome!

[@fredsharp7419]

Your enthusiasm for mathematics is infectious and your explanations are wonderfully clear! Keep up the excellent work!

[@divyoroy9056]

Blackpenredpen: Does everything in one take Me: We already know you can do it

[@blackpenredpen]

Divya Roy hahah thanks!

[@oliverguidetti7569]

God almighty bless this man. Bless this channel. I am a mathematician that grew up on KZbin lessons. I WISH WITH ALL MY HEART THIS WAS AROUND WHEN I WAS A YOUNG MATHEMATICIAN.

[@Samir-zb3xk]

2:10:24 you can derive the identity cos(x) = 1/√2 [cos(x-π/4) - sin(x-π/4)] by expanding cos(x+π/4) using the angle sum identity then subtracting π/4 from all angles

[@ffggddss]

Nice! And carefully explained! Another possible refinement would be to re-index each sum to make x appear without additive exponents: ∑₀⁰⁰4ⁿxⁿ⁺¹ = ∑₁⁰⁰4ⁿ⁻¹xⁿ ∑₀⁰⁰(-1)ⁿ/9ⁿ⁺¹x²ⁿ⁺⁴ = ∑₂⁰⁰(-1)ⁿ/9ⁿ⁻¹ x²ⁿ etc. But this is largely a matter of taste. Fred PS. Speaking of 26.2, the play length of this video happens to be very close to my time for my very 1st marathon, which I just marked the 40th anniversary of on Monday, 11/4 ! A few running math trivia: • The official distance for the marathon, before it was metricized, was 26 mi 385 yd = 26 mi 70 rod = 26 mi 1¾ furlong = 26 ⁷/₃₂ mi = (839/32) mi, making 839 the "marathon prime." • The newer, metric standard = 42.195 km = 42,195 m, is shorter than the English standard by exactly 1.2 cm

[@blackpenredpen]

Ahhh thank you! And running a mile around 3:36 is so good! Thank you for the math fun facts about the marathon too! My goal is to run a marathon under 4 hours! My best time so far is about 4:22 but I did that back in 2007. And Fred, did you see Eliud who ran 26.2 miles under 2 hours last month? It was insanely impressive!!

[@ffggddss]

​@@blackpenredpen Yeah, that was fantastic! I never thought 2 hours would be broken. I was amazed back in the 70's, at times under 2:10 (5 min/mi is about 2:11). And personally, I'm kinda sad that this takes the marathon best-time out of my "hour category" (my best was just barely under 3 hr; markedly better than my other 9 tries - I like to say that I was born with one good marathon in me, and I ran it in 1981). Nowadays in a marathon, I'd have to walk, and probably couldn't break 7 hours (that's roughly 10 min/km). "... a mile around 3:36 is so good!" - Yeah, that would be a world record!! OK, I know you meant marathon, and hours, not minutes ;-) As for my early marathons - that distance was never my favorite; I used to do a lot of 5 km, 5mi, 10km, 15 km, 10 mi, half-marathon - those sorts of distances. And training for the marathon would keep me sharp for those shorter distance races. My training pace was typically 7½ to 8 min/mi; sometimes a little faster or slower, depending on distance; and I benefitted from training regularly at club training sessions, where I could train with a group that was just a bit faster than I - not too much, or I'd exhaust myself before getting any real improvement. Near the end of my road-racing days, my marathon times were a bit over 4 hr. I had to give up distance running in the late 90's, due to the way it aggravated my knees. I try to do lots of biking and fast walking nowadays. Best of luck with your 4-hr goal - train smart, and be confident in your ability to overcome that hurdle. I'm sure you'll do it one of these times. Fred

[@mtrichie111]

Thanks Professor, I have many friends, but this friend is the only friend I can really trust!

[@blackpenredpen]

Hahahaha glad to hear!!

[@robertforster8984]

All my friends are in heaven.

[@mtrichie111]

All mine were sent to the other place

[@dogcat4572]

@@blackpenredpen I’m

[@dogcat4572]

@@blackpenredpen I’m Zip jgo but

[@JohnSmith-rf1tx]

For Q18. Power Series of cos(x) at a=pi/4, 2:09:54, or other trig functions centered at not nice values, you asked for a way to do it without using Taylor's formula. You can use the angle sum/difference identities for cosine and sine to rewrite cos(x) in a workable form that only contains (x-pi/4) terms instead of x. You have to use both the cosine and sine angle sum/difference identities in order to make it work, but it's nicer than it looks at first glance since you know the actual values for cos(pi/4) or sin(pi/4). And in this particular example, you get some tan(pi/4) which equals 1, and therefore allows some simplification. I get cos(x)=(1/sqrt2)*[cos(x-pi/4)-sin(x-pi/4)].

[@aidtheneedy4640]

Am glad I came across this just right before my calc 2 exams. Thanks a lot !

[@VibingMath]

OMG Whoaa one-take video again!!!!!! Your Cal 2 students are so good and proud to have you as the teacher! Great job man and I hope your students can get good results in tests and exams!

[@blackpenredpen]

Mak Vinci awww thank you for your nice comment, as always!

[@VibingMath]

@@blackpenredpen Yay!

[@anuragguptamr.i.i.t.2329]

@@blackpenredpen Please make a video on Maclaurian power series expansion of e^sinx upto 6 terms at least.

[@vasilisr7]

The Best Math Teacher in the World! Thank you for everything you learn to me!

[@drpeyam]

26.2 power series, but 100% fun!

[@blackpenredpen]

Dr Peyam definitely!

[@Vandarte_translator]

Right now this is what I am studying in my math course. What a coincidence! THANKS!

[@FelipeRuedaH]

For cos(x) at a=pi/4 you could just do cos(x) = cos((x-a)+a) =cos(a)cos(x-a)-sin(a)sin(x-a)

[@blackpenredpen]

Felipe Rueda ahhhh I didn’t think of they

[@doodelay]

I've realized that I can't use your marathon vids to teach me what i don't know yet, but they are INCREDIBLY useful for sharpening intuition once you've gone through the basics!

[@sonicmaths8285]

in maths theory should always come first

[@lassaadbelhiba8649]

Many thanks Professor from Tunisia. 3 hours went by quickly

[@yourockero]

I love you Man. Greetings from Colombia. South America.

[@violintegral]

For cosx at a=π/4, you can also use the angle sum identify for cosine: cos(a+b)=cos(a)cos(b)-sin(a)sin(b) to write cos(π/4+πn/2) as 1/√2(cos(πn/2)-sin(πn/2)). But the answer bprp has is fine.

[@violintegral]

For number 23, you can also find the Taylor polynomial expansion of 2x³-5x²+1 using a form of undetermined coefficients, like we do for partial fraction decomposition in integrals. You can set 2x³-5x²+1=A(x-1)³+B(x-1)²+C(x-1)+D, and then solve for constants A, B, C, and D by equating the coefficients of each power of x on the left and right side, then solve a simple system of equations. I got the same answer as bprp did with his method using Taylor's Formula. That is, 2x³-5x²+1=2(x-1)³+(x-1)²-4(x-1)-2.

[@skwbusaidi]

For question 3 We can also do long division (2x+1)/(1+x) = -2 + 3/(1-x) = -2 + sum of 3x^n from n=0 We can get the same result if we leave the first from the series

[@ThePeterDislikeShow]

My favorite Calculus subject! I love calculating famous constants to billions of digits using such!

[@NinjaMartin]

These study guides are amazing, Sir! I mean, they are the perfect companion in preparation for exams! Great work, BRRP!

[@kinama33kinama]

I love Maths , but this guy makes me laugh while doing maths , he is great.

[@khalidheyredin1414]

yeah

[@twosongs7396]

I don’t know what your major is (I’m guessing it’s Math) but please do keep Teaching in mind as you EXCEL as a Teacher and I much appreciate this video, being a new follower of yours.

[@blackpenredpen]

Thank you 😃

[@sleepsy1522]

He is not in college he probably graduated in like 2014-2017

[@TheAbelPower]

Mesmo eu sendo brasileiro, esses vídeos me ajudam demais, pois os vídeos br só resolvem os problemas mais triviais e só aqui eu vejo problemas que realmente são cobrados em um curso de graduação

[@davibatista8650]

Sim.

[@BootesVoidPointer]

Power series is exactly the subject where I'm currently at in my self study of calculus 2, so this video is extremely helpful. Thank you!!

[@fyrerayne8882]

You’re walking a fine line between wizardry and sorcery!

[@nargesafzali-r1i]

oh my god... you are amazing teacher !please keep going

[@violintegral]

A much better way to find the power series for artanh(x) is to recognize that the antiderivative of 1/(1-x²) is artanh(x)+C. Then you can use the best friend to write 1/(1-x²) as the sum from n=0 to ∞ of x^(2n). Next, integrate x^(2n), which gives x^(2n+1)/(2n+1). So artanh(x) is equal to C plus the sum from n=0 to ∞ of x^(2n+1)/(2n+1). Use the definition of artanh(x), artanh(x)=1/2ln((1+x)/(1-x)) to find C easily by plugging x=0, so C=0. So artanh(x) is equal to the sum from n=0 to ∞ of x^(2n+1)/(2n+1), R=1, I=(-1,1). You can easily check the endpoints of the interval of convergence using the limit comparison test with the harmonic series, or by using the definition of artanh(x), as bprp showed.

[@kirbo722]

Very helpful video!!!!!! Thank you for making this ^^

[@marceloguzman646]

my true calc 2 teacher ❤️

[@alejrandom6592]

omg I love this channel :) I like math but learning from you make me love it

[@chirayu_jain]

Again a one take video 😱, they are really hard to do and you have also written the time stamps 😱. Can you do 100 limits next time 🙏

[@blackpenredpen]

I have been planning on that already. : )

[@chirayu_jain]

@@blackpenredpen looking forward for it

[@nisargbhavsar25]

How are you chirayu ?

[@chirayu_jain]

@@nisargbhavsar25 Fine

[@hinabhavsar1730]

@@chirayu_jain Great

[@gamingletsplays2518]

Hi again! I'm gonna assume it's a good video even though I haven't watched yet:-)

[@gamingletsplays2518]

Also you should do a q & a or another one if you already have.

[@klausolekristiansen2960]

Safe bet.

[@rob876]

You make it all look so simple. You have exceptional teaching skills. I'd like to see your explanation of how to solve a cubic equation. All the videos I've seen so far are so badly explained. Maybe this wouldn't be the requirement of any syllabus but it would be a fascinating video to watch.

[@maanavgurubaxani4166]

It is very helpful!! I would request to make this 24 question series on more and more calculus topics as it makes our topic very clear!! Thanks for the video

[@ieqvilamaria619]

Bprp I wanna say that your videos are incredible and they are doing I love math more and more hahaa

[@blackpenredpen]

Thank you : )

[@kianmath71]

For question 11 you could also use the ratio test to get 0 for thenlimit which shows the interval of convergence is from - infinity to infinity

[@younesabid5481]

Could you please do a video on the inverse functions of Tetration 😊

[@blackpenredpen]

Ok!

[@hatacoyama1246]

i like this

[@randomdude9135]

What's that?

[@hatacoyama1246]

Random Dude its repeated exponentiation, like how multiplication is repeated addition, and exponentiation is repeated multiplication. its the next step up from exponentiation.

[@randomdude9135]

@@hatacoyama1246 Oh, like the one used in Graham's number?

[@sunimod1895]

I literally have a test on this tomorrow. Thanks!

[@blackpenredpen]

Perfect!

[@skltfz4997]

is something wrong at Q.22 ? 2:45:03, it should be (-1)^n instead of (-1)^n+1

[@VortexGamesYT]

Very cool video, bprp. Please do 100 limits in one take video!

[@Tom-xq7ys]

Sir YOU ARE THE MAN I WANT TO SOLVE MY MATH TEST

[@risheraghavendira6042]

Lol😂😂 nice advertising 1:45:43.... Btw, I'm 14 and I love your vids even though I am not in calc Appreciation from India

[@matt-thegeek6486]

You are to Good mate continue like that!!!!!!!

[@holyshmowcow]

Just started power series last week so this was a cool video

[@ThAlEdison]

cos(x)=cos((x-pi/4)+pi/4) =cos(x-pi/4)/sqrt(2)-sin(x-pi/4)/sqrt(2) But I think Taylor series may be faster to simplify.

[@beachwave5705]

quick note: dont take 4 summer classes. with that being said, I got my bio and calc 2 finals this Wednesday and Thursday. mind you my calc 2 course lasted 10 weeks, although really it was only 9. bio is 6 weeks. if I pass these classes, 25% of the credit goes to u. other KZbinrs were organic chem tutor, amoeba sisters, Patrick jmt, Nancy pi, ms shaw, LUDUS, but I always finish a topic with one of your marathon videos. my top favorite of your marathons or atleast really long videos were both disc/washer and shell method, series, and this video.

[@marinecwo4]

AWESOME! Great endurance and very inspiring! You are a champion!🏆🥇

[@Mario_Altare]

It's wonderful! (quot. Paolo Conte) A suggestion for a possible next video: Improper integrals marathon/ultimate study guide) with integrals such as "ln x dx/(x^2+1)", "dx/(x^4+1)", etc. It would be as great as the other marathon/ultimate-study-guide videos :-)

[@lipelego22]

Watching this at 3 am for my calc 2 final test lol Much love from Brazil at Unesp(São Paulo state university), you and 3blue1brown help me a lot in calculus

[@violintegral]

For question 26, I found that the interval of convergence was (-1,1). I checked the endpoints using the ratio test on the sum from n=0 to ∞ of (-1/2 choose n)(-1)^n/(2n+1). The limit diverged to infinity, so both endpoints -1 and 1 weren't included in the interval of convergence.

[@landerzuluetaloizaga342]

I love this guy so much

[@onurdemir353]

Thank you so much .Your videos make me math love.

[@davideventurini9699]

Thanks! Really useful since I'm studying these great part of mathematics!

[@ГормонСвободы]

🙏 Thank You, Sansei!

[@elfabri666]

THANKS YOU SAVED MY LIFE

[@ccd30132]

If you put n=0 on the bottom at 30:14, you would have to add something to 0^(-1)

[@merbst]

Congrats on running in the Long Beach Marathon.

[@blackpenredpen]

Thank you!!!!

[@rodwayworkor9202]

Hey BPRP. Your videos have amazed me for ages, yet I believe that you haven't yet shown your true capability yet. How about, in one video, you solve some amazing Olympiad problems suggested by the comment section, for long, long time? Or maybe solve all of them in 1 minute :D

[@WolfgangKais2]

You could have used „partial fraction decomposition“ in problem 3: (1+2x)/(1-x) = 3/(1-x) - 2.

[@losiu998]

0:25 where can I get this paper?

[@blackpenredpen]

losiu998 hi there, it’s in the description

[@sergiolucas38]

Great video, much obliged.

[@nicolassaade576]

bro...absolute legend.

[@iamironman5442]

It's very useful...👍

[@migo77]

Could you please share that paper at 0:24 ?

[@artichilli]

😍😍🤩🥰 thank you sirrr

[@ynogomez]

please, give me the link for 0:30

[@TheNachoesuncapo]

Yes,HE-IS-SERIOUS.

[@gagikandjergholi5154]

thanksssssssssssssssssss alotttttt, it was really fun

[@roflmfao4life]

Ooh you should totally do a follow up video for double factorial

[@yassine321]

could anyone send me the link for the sheet at 0:28 ??

[@yassine321]

i just download it from scribd c:

[@MeleseTesfaye-i3k]

You are the greatest

[@SrPuma-df7zp]

Thank U so much!

[@blackpenredpen]

My pleasure! : )

[@grantwiesehan9515]

thanks so much! this helped a lot!

[@akamsgd5393]

When school gets hard I come here because it really prepares me for what I need to know

[@tahraf2308]

fell asleep while watching youtube, auto play gave me this gem

[@-.SkyArt.-]

0:24 PLEASEEE can you upload the file for that sheet? I REALLY need it! Thanks!

[@skwbusaidi]

No 21 It is easier to use the fact that integeral of 1/(1-x^2) = tanh^-1 x +c

[@hiradt5011]

Thank you sir, this really helped!!!

[@subhrajitroy1477]

Sir, Where can I share mind boggling questions with you? Is there any platform except Twitter? Because I don't have an account there. I am an Indian High School student preparing for the infamous JEE Advanced exam ;)....But I am highly interested in Maths, and one of your loyal subscribers

[@timtaler2435]

Read Henri Cartan's book "Elementary Theory of Analytic Functions of One or Several Complex Variables" or even a bit advanced "Analytische Stellenalgebren" from Grauert, Remmert, insted of having a gazilllion examples. You will know much more interesting about power sreries.

[@onlineskillsbyimranbilal]

U r the great teacher

[@skwbusaidi]

For 19 and 20 we can also use Sinhx = -i sin(ix) Coshx = cos(ix)

[@sophiabnrm8730]

You effort is insane. I really wish you best of luck and success. Do you plan any videos on graph theory proofs as I am having trouble with some of them and cant seem to find helpful tutorials. Thanks in advance!

[@blackburd]

I never knew that a magic 8 ball could be used as a microphone! :)

[@alejrandom6592]

44:38 nice that alternating harmonic series converges to ln2

[@crappymccrappen4897]

*easy fix* You should tell everyone about the fix you are using. We will become better at math.

[@jarikosonen4079]

6.) Ln(0)=-Inf might work similarly with the power series... (both ln()&power series 'diverge', but in same manner). Then why not included like x=[-1,1]? What is series: 1+x^2+2/3*x^4+17/45*x^6+... How to convert it to trigonometric if it is?

[@carlosgiovanardi8197]

i need i guy like you to teach me automation!!

[@darrenho1732]

@3:03:28 What if r is not a natural number?

[@blackpenredpen]

Still works!

[@charleswoodard8478]

Q16: Desmos does not concede the -1 external multiplicand external to the sum. F.e. -1(sig.) an

[@shivamhanda1748]

Hi where can i find the sheet you showed us at the start of the video to test whether a series converges or not?

[@Bitinful]

@blackpenredpen could u tell me what medal are u wearing on the photo?