Example 1.6 Find the closed form fu

Example 1.6

Find the closed form function for the following series:
1 1 1 1
f (x) = 1 + x − x2 − x3 + x4 + x5 + · · · (1.23)
2! 3! 4! 5!
The signs of the terms are positive for the first two terms then negative
for the next two terms and so on. The series should be grouped as
1 1 1 1
2 4 3 5
f (x) = (1 − x + x + · · · ) + (x − x + x + · · · ), (1.24)
2! 4! 3! 5!
that will converge to
f (x) = cos x + sin x.