For x>0, the power series 1−3!x2+5!x4−7!x6+...+(−1)n(2n+1)!x2n+... converges to which of the following?
Note that only the even powers are present and the coefficients alternate. This implies a sine or cosine function.
Normally, the expansion of cosine is:
1−2!x2+4!x4+...
The expansion of sine is:
x−3!x3+5!x5+...
Neither of these fit, but if we divide each term of the Taylor expansion of sine by x, we obtain the give series.
xx−3!x3+5!x5+...
=1−3!x2+5!x4+...