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

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
$$ 1 + \underbrace{\frac12+ \frac12}_{2\text{ halves}} +
\underbrace{\frac13 + \frac13+ \frac13}_{3\text{ thirds}} +
\underbrace{\frac14 + \frac14 + \frac14+  \frac14}_{4\text{ fourths}}+
\underbrace{\frac15 + \frac15+  \frac15+ \frac15+\frac15}_{5\text{ fifths}}+
\cdots $$

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