Quote:

Taylor Series expansion of y = sin(x + cos(x)) at x = Pi/2 (up to the x^{7} term, say)

While it's easy to get the result with a calculator that is able to perform symbolic differentiation I'm trying to show that this isn't needed. The arrays contain the first coefficients of the Taylor Series at x = Pi/2.

**x** = [Pi/2, 1, 0, 0, 0, 0, 0, 0]

**cos x** = [0, -1, 0, 1/6, 0, -1/120, 0, 1/5040]

Thus:

**u = x + cos x** = [Pi/2, 0, 0, 1/6, 0, -1/120, 0, 1/5040]
**sin u** = [1, 0, -1/2, 0, 1/24, 0, -1/720, 0]

**v = u - Pi/2** = [0, 0, 0, 1/6, 0, -1/120, 0, 1/5040]

**v**^{2} = [0, 0, 0, 0, 0, 0, 1/36, 0, -1/360]

We only have to take the 2nd coefficient (-1/2) of *sin u* into consideration, since v^{4} is O(x^{12}) already.

So we end up with:

**sin(x + cos x)** = 1 - v^{2}/2 + ...

= 1 - (x - Pi/2)^{6}/72 + (x - Pi/2)^{8}/720 + ...

All we need are operations that interpret an array correctly as a Taylor Series. This shouldn't be too difficult to implement. Though I have no idea whether this would be possible within the WP-34s project, it's far from symbolic differentiation. Maybe as an idea for a follow-up project then?

Kind regards

Thomas

*Edited: 21 Jan 2013, 2:36 a.m. *