im looking to understand the tangent taylor series, but im struggling to understand how to use long division to divide one series (sine) into the other (cosine). I also can't find examples of the Tangent series much beyond X^5 (wikipedia and youtube videos both stop at the second or third term), which is not enough for me to see any pattern. (x^3/3 + 2x^5/15 tells me nothing).
Wiki says Bernouli Numbers which i plan on studying next, but seriously, i could really use an example of tangent series out to 5-6 just to get a ballpark of what's going on before i start plug and pray. If someone can explain why the long division of the series spits out x^3/3 instead of x^3/3x^2, that would help too,
because I took x^3/6 divided by x^2/2 and got 2x^3/6x^2, following the logic that 4/2 divided by 3/5 = 2/0.6 or 20/6. So I multiplied my top and bottom terms for the numerator, and my two middle terms for the denominator (4x5)/(2x3) = correct.
But when i do that with terms in the taylor series I'm doing something wrong. does that first x from sine divided by that first 1 from cosine have anything to do with it?