Hi there! look, I have an existencial doubt haha. The other day i was working on the taylor series for the arctan(x), after a few derivations it was so insane to do. Then i decided to do a research on my Spivak and i found that there is a very simple way to do that kind of series. They say that 1/(1+x^2)= 1-x^2+x^4-x^6+...+(-1)^n.x^(2n)+((-1)^(n+1).x^(2n+2))/(1+x^2). Notice that it is the derivate of the arctan(x), then if you integrate on both sides... you'll get the taylor series for the arctan(x). That was-such-a-BIG-discover , you can try to prove it by multiplying by (1+x^2) on both sides.
Hello, slightly off topic but i just need to ask something about the fibonacci sequence: we have x(n)=f[x(n-1),x(n-2)], and we also have x(n)=g(n). But is there, or is it useful to have x(n)=k[(x(n-1),n]? (I mean the nth term as a function of the previous term and the index). I have been out of the math world for a few years, so i wouldn’t know. Btw, love your channel and content. Keep up the good work.
BlackpenRedpen and now possibly BlackpenRedpenBluepen - Love your work and your attention to detail, you are really talented and appear to be an excellent teacher. But wasn't your proof just an exercise in complimentary angles and teaching rigor and patience with negatives. A simpler proof for Sin(A)Sin(B) is just the expansion of Cos(A+B) subtracted from the expansion of Cos(A-B) and then divide both sides by 2?
I love how many creative ways there are to manipulate trig functions, it makes working with them a puzzle
I like how you change the way you write down the answers over time
Ayyyyyy my dude back at it again with the quality proof
Hi there! look, I have an existencial doubt haha. The other day i was working on the taylor series for the arctan(x), after a few derivations it was so insane to do. Then i decided to do a research on my Spivak and i found that there is a very simple way to do that kind of series. They say that 1/(1+x^2)= 1-x^2+x^4-x^6+...+(-1)^n.x^(2n)+((-1)^(n+1).x^(2n+2))/(1+x^2). Notice that it is the derivate of the arctan(x), then if you integrate on both sides... you'll get the taylor series for the arctan(x). That was-such-a-BIG-discover , you can try to prove it by multiplying by (1+x^2) on both sides.
I was excited about the possibility of a live feed today, but tomorrow is another day :-)
Factoring out factored out factors.
Where is your blackbord and chalks or the whiteboard and famous blackpenredpens???
wow I never thought about that before!!!
Such a lovely ending :)
Hello, slightly off topic but i just need to ask something about the fibonacci sequence: we have x(n)=f[x(n-1),x(n-2)], and we also have x(n)=g(n). But is there, or is it useful to have x(n)=k[(x(n-1),n]? (I mean the nth term as a function of the previous term and the index). I have been out of the math world for a few years, so i wouldn’t know. Btw, love your channel and content. Keep up the good work.
Can you make videos explaining how to solve equations involving the floor function? An example of such equation would be floor(x)-2floor(x/2) = 1.
Yay!
beautiful are, trigonometric proofs
Sir Please make a video at Coordinate Geometry.
Curious why you haven't done a video on the "magic bullet" substitution that turns any trig integral into an algebraic.
Find equation of circle with center on a line y= - x has radius 4 and passes through origin?
Hey blackpenredpen
BlackpenRedpen and now possibly BlackpenRedpenBluepen - Love your work and your attention to detail, you are really talented and appear to be an excellent teacher. But wasn't your proof just an exercise in complimentary angles and teaching rigor and patience with negatives. A simpler proof for Sin(A)Sin(B) is just the expansion of Cos(A+B) subtracted from the expansion of Cos(A-B) and then divide both sides by 2?
❤
Interesting how sin is and odd function and therefore the sign of its arguments matter (I'm enjoying the sin v/s sign duality here).