So the question is from a textbook. I have spent some time thinking about this, but I do not think I have an adequate answer. It is the following:
Let $a_n$ and $b_n$ be two sequences such that the series $\sum^{\inf}_{n=1} a_n$ converges and $\lim_{n->\inf}\frac{a_n}{b_n} = 1$
a) Prove that the series $\sum^{\inf}_{n=1} b_n$ converges.
b) Prove that the Fourier series $\sum^{\inf}_{n=1} (a_n cosn x+ b_n sinnx)$ converges unformly and absolutely on $R$.
The problem is that I am stumped on the first part. While it seems intuitively obvious that as n approaches infinity, $b_n$ would have to converge in order for the fraction $\frac{a_n}{b_n}$ to converge, but how would one go about proving this rigorously?
As for the second part, is the answer a corollary to answering the first one? Any help would be much appreciated.