Some examples of Taylor series:
ex = 1 + x + (1/2!) x2 + (1/3!) x3 + (1/4!) x4 + . . .
= Σ[n = 0, ∞, (1/n!) xn]
sin x = x - (1/3!) x3 + (1/5!) x5 - (1/7!) x7 + . . .
= Σ[n = 0, ∞, (-1)n (1/(2n + 1)!) x2n + 1)]
cos x = 1 - (1/2!) x2 + (1/4!) x4 - (1/6!) x6 + . . .
= Σ[n = 0, ∞, (-1)n (1/(2n)!) x2n]
sinh x = x + (1/3!) x3 + (1/5!) x5 + (1/7!) x7 + . . .
= Σ[n = 0, ∞, (1/(2n + 1)!) x2n + 1]
cosh x = 1 + (1/2!) x2 + (1/4!) x4 + (1/6!) x6 + . . .
= Σ[n = 0, ∞, (1/(2n)!) x2n]
ln x = (x - 1) - (1/2) (x - 1)2 + (1/3) (x - 1)3 - (1/4) (x - 1)4+ . . .
= Σ[n = 1, ∞, ((-1)^n) (1/n) (x - 1)n]
1/(1 - x) = 1 + x + x2 + x3 + x4 + . . .
= Σ[n = 0, ∞, xn] (a geometric series)
tan-1 x = x - (1/3) x3 + (1/5) x5 - (1/7) x7 + . . .
= Σ[n = 0, ∞, (-1)n (1/(2n + 1)) x2n + 1]
These series can be used to find some rather strange patterns in otherwise normal functions, like Euler's equation (eiπ = -1). Some other weird properties are the facts that cos ix = cosh x, and sin ix = i sinh x.