Taylor series for $\sec(x)$

Question

I'm trying to compute the Taylor series for $\sec(x)$ but the derivatives are getting way too complicated. So how do I go about this without having to calculate all the derivatives? I tried to build some kind of relationship with the series for $\cos(x)$ but I didn't get anything meaningful.

Answer 1

Look at the Boustrophedon table: $$\matrix{1\\0&1\\1&1&0\\0&1&2&2\\5&5&4&2&0\\0&5&10&14&16&16\\61&61&56&46&32&16&0\\0&61&122&178&224&256&272&272\\1385&1385&1324&1202&1024&800&544&272&0}$$ etc. Each row is the series of partial sums of the previous row, but at each stage one reverses the order we add up and enter the partial sums. Any, from the first column we read off $$\sec x=1+\frac{x^2}{2!}+\frac{5x^4}{4!}+\frac{61x^6}{6!}+\frac{1385x^8}{8!}+\cdots.$$ The right-most elements also give $$\tan x=x+\frac{2x^3}{3!}+\frac{16x^5}{5!}+\frac{272x^7}{7!}+\cdots.$$

There's a good discussion on this in Concrete Mathematics by Graham, Knuth and Patashnik.

Answer 2

There is the expression $$\sec z = 1 + {z^2\over 2} + {5z^4\over 24} + \cdots + {(-1)^n E_{2n}\over (2n)!} z^{2n} + \cdots$$ for the Taylor series, where $E_{2n}$ is an Euler number; see Abramowitz and Stegun, {\sl Handbook of Mathematical Functions}, p.~75, Equation 4.3.69; and the discussion of Euler numbers on pp.~804--805, and the table of values for Euler numbers on p.~810. I don't know if there is an efficient way of computing Euler numbers for a large index, but it is at least a different place to start. (I stumbled on this looking for a proof that all the Taylor coefficients of~$\sec(z)$ are positive.)