Define the sequence (cn) and (sn) inductively by c1(x) := 1, s1(x):= x, and

for all n ˆˆ N, x ˆˆ R. Reason as in the proof of Theorem 8.4.1 to conclude that there exist functions c : R †’ R and s : R †’ R such that (j) cʹʹ(x) = c(x) and sʹʹ(x) = s(x) for all x ˆˆ R, and (jj) c(0) = 1, cʹ(0) = 0 and s(0) = 0, sʹ(0) = 1. Moreover, cʹ(x) = s(x) and sʹ(x) = c(x) for all x ˆˆ R.

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Cn(l)dt, C+1 0