Chapter 9.5, Problem 55ES

(a) We will see later that the polynomial $1-x^2/2$ is the “local quadratic� approximation for $\cos x\text{at}x=0.$ Make a conjecture about the convergence of the series ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\left[1-\cos \left(\frac{1}{k}\right)\right]}$ by considering this approximation.

(b) Try to confirm your conjecture using the limit comparison test