Q264, power series expansion of a rational function

[@BigDBrian]

if x is 0 then the expression for the condition of x reads: lol

[@blackpenredpen]

lol! I like this!

[@whiz8569]

I wanted to give 5 lol's to this, but, alas, I can only give less than 3.

[@ashbuchanan11]

I used to wonder how some people have so much fun with math, but the smirk on this guy's face every time he gets to an interesting conclusion has pretty much answered that question.

[@tinius7mosvold807]

The clearest I've ever had it explained to me, and all with a big smile. I am unable to verbally convey how much I enjoyed this video. Thank you

[@benjaminfranklin1081]

I seriously appreciate how much work you put into these videos, from explaining each step, in depth explanations and problems, and just actually doing your best and enjoying the math. Thank you for these amazingly helpful videos, and your kind spirit, keep it up, you are doing a great job!!!

[@blackpenredpen]

Thank you Benjamin and I will continue to work hard!

[@alvaroperezrivera5069]

Im speak spanish but i understand the video No doubt, maths its a universal language

[@rubentuquerrez]

Omg, You're awesome I just find your channel and now I'm a big "fan" of you. I'm gonna watch all your videos in this week

[@blackpenredpen]

Rubén Túquerrez thank you!

[@antoniomora1621]

"Using My Best Friend", I do that on a daily basis

[@aviralaryal7332]

hol' up

[@NisargJain]

I might be your best friend

[@MG-hi9sh]

Lol, time to change that then.

[@cristianmorales5207]

I had to take a semester off because I failed Differential Equations and Linear Algebra twice and I felt like I was a huge failure but you make this stuff seem so easy. Thanks

[@kostagerosky534]

Hello from Macedonia, I simply love your videos!!! :)

[@simpletn]

Overall my favourite channel.

[@DaanSnqn]

It's sad that my best friend is 1/(1-x) instead of real person. 😢

[@MG-hi9sh]

I know how it feels.

[@davedonnie6425]

You're know you're lonely when this is your best friend.

[@blackpenredpen]

My best friends are my viewers : )

[@strangealiengodofpower7628]

Could you do divergent series plzzz...

[@AlgyCuber]

5:08 you could have said x^2 is always positive or zero and you don't need the abs value? x^2 < 9 |x| < 3

[@oshii4187]

best bread best rent bedspread Source: yt automatic captions

[@peteneville698]

Is " |x.x| = |x|.|x| " more of a definition/rule kind of thing, or does it technically need a proof of its own?

[@ianmoseley9910]

Pete Neville I was thinking the same thing - sure there must be a proof

[@frede1905]

Ian moseley The absolute value of a number x is defined as sqrt. (x^2), when we only take the positive value of the square root. So let's say that x is the product of two numbers, let's say a and b. So we have: abs. (x)=abs. (a•b)=sqrt. ((a•b)^2)=sqrt. (a^2•b^2)=sqrt. (a^2)•sqrt. (b^2)=abs. (a)•abs. (b). So the absolute value of a•b is the absolute value of a times the absolute value of b.

[@leonardobarrera2816]

thanks

[@ruinenlust_]

great video!

[@vineethreddy.s]

Excellent bro...

[@pietroberardigili562]

Do some videos about cool definite integrals

[@blackpenredpen]

pietro berardi gili hmm.... I will try when I have a chance during the summer

[@blackpenredpen]

pietro berardi gili i just uploaded one!

[@CTJ2619]

I may have missed this in my Math class but how are you able to multiply the 2x^5/9 by the term on the rhs of the summation. What is the principle that allows you to multiply across a summation (See 4:00)

[@optimalzeeeee]

If you just think about it the same way you would a finite sum, you can see why. For example, n+nx+nx^2+nx^3... = n(x+x^2+x^3...) in the same way that 3x^2+3x = 3x(x+1).

[@holhol4283]

Sir could you please make a video solving the integral secx/cos2x

[@mohanlalchoudhary872]

How many best friends do you have?

[@blackpenredpen]

Science with LalitKumar 1

[@07bmarshall]

What is our best friend? Thank you.

[@canonpi]

how is this simplification???

[@Daniel-Brous]

What's the radius of convergence used for? I'm curious

[@blackpenredpen]

So that we know what x values we can legitimately plug in to the power series. For example, I can plug x=2.9 into the series that I got in the video and get Series of 2(-1)^n*(2.9)^(2n+5)/9^(n+1) and I know that will converge to 2(2.9)^5/(9+(2.9)^2) since x=2.9 is on I. On the contrary, I cannot plug in x=3.1 for example.

[@Daniel-Brous]

Okay. So then what's the use of the power series? Doesn't it just limit our ability to evaluate the given expression?

[@TheYoshi463]

+Daniel Brous Often a power series is more easy to deal with (especially for functions like sinx etc).

[@not_vinkami]

I was just thinking how to integrate it

[@MDNayem-qc3bu]

Love

[@saxbend]

why do you use the mods (absolute value) on the x^2 term? Are we allowing x to take complex values?

[@jordanberman6194]

Can you please do a proof for the best friend? I would really love to see! Love your vids xxx

[@blackpenredpen]

Jordan Berman i did. It's at the end of the video.

[@blackpenredpen]

Jordan Berman there's a link

[@PerfectlyCrafted]

can't you just use the quotient rule?

[@blackpenredpen]

?

[@PerfectlyCrafted]

blackpenredpen nvm I'm stupid sorry

[@mariomario-ih6mn]

@@PerfectlyCrafted What did you mean

[@PerfectlyCrafted]

@@mariomario-ih6mn I was learning derivatives at the time and didnt know the difference. I was very dumb hahha

[@mariomario-ih6mn]

@@PerfectlyCrafted You're not dumb if you make a few mistakes

[@Headerman300]

Lets look at this S 😂😂

[@sansamman4619]

4:15 . Why didn't you say 'x' to the 7n power.7n=2n+5 ?

[@sansamman4619]

ByTheWay i can't believe you,you are a monster you use your best friends you just have them to use them you kill your best friends every video(by erasing them ) :(

[@nilswitt6314]

Since 7n ≠ 2n+5. Proof by contradiction: Assume: 7n = 2n+5. 7n = 2n+5 |-2n 5n = 5, therefore n=1, but n cannot equal to a constant number, since it is a variable. It can also be seen that 7n ≠ 2n+5 if n≠1 is plugged into the equation

[@ezu8501]

you a beautiful man.