Taylor and Mcclaurin Series

When forming the sequences of the polynomial stuff are there any tips or tricks to forming it, or do I just have to be gifted in noticing patterns? What techniques do you use?

You must be gifted in studying what mathematicians have done after Maclaurin and Taylor :-)

I mean: you haven't to invent the sand. Like in programming languages, there is a theory and there is an 'art'

suppose you have to Maclaurin f(x) = sin x

let's imagine f(x) like f^0(x)
first derivative as f^1 and so on

f^0 = sin
f^1 = cos
f^2 = - sin
f^3 = - cos
f^4 = sin

there is a "periodicity" that everybody can see

To say nothing of e^x

take f(x) = log(1 + x)

f^0 = log(1 + x)
f^1 = 1/(1+x)
f^2 = -1/(1+x)^2 = - (1+x)^(-2)
f^3 = - (-2)(1 + x)^(-3) = 2(1 + x)^(-3)
f^4 = - 3·2 (1 + x)^(-4)

I bet my hat that

f^n = ( - 1)^(n-1) (n - 1)! (1 + x)^(-n)

so building the series is much easier

(f^n(0) x^n)/n!

log(1+x) = series from n=1 to infinity ((-1)^(n-1) x^n)/n

log 2 = log(1 + 1) = 1 - 1/2 + 1/3 - 1/4 ...

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