Does the series for $\cos(x)/x$ converges?

Question

The sequence of $$ a_x ={\cos (x)\over x} $$ does converge to zero. As a result, intuitively $$ \sum_{x=1}^\infty {\cos (x)\over x} $$ should also converge right? But I've been told that the series diverges. This shouldn't be true... right?

Answer 1

It is not at all intuitive to me that the series ought to converge simply because the terms go to zero. For example $\sum_{n=1}^{\infty} \frac1n$ is well known to diverge even though $\frac1n\to 0$. Or, as an even easier example, consider $$ 1 + \underbrace{\frac12+ \frac12}_{2\text{ halves}} + \underbrace{\frac13 + \frac13+ \frac13}_{3\text{ thirds}} + \underbrace{\frac14 + \frac14 + \frac14+ \frac14}_{4\text{ fourths}}+ \underbrace{\frac15 + \frac15+ \frac15+ \frac15+\frac15}_{5\text{ fifths}}+ \cdots $$

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It does look like your particular series converges (conditionally), by Dirichlet's test, though.