# Does the series for $\cos(x)/x$ converges? [duplicate]

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?

• What is $n$ in your definition of $a_n$ ? Do you mean $cos(n)/n$? Aug 19, 2018 at 11:40
• Are you sure you formatted the question right? It isn't making much sense to me at the moment. Aug 19, 2018 at 11:41
• sorry, i'll edit it Aug 19, 2018 at 11:41
• When $n=0,$ the term $\cos(0)/0$ is not defined. Do you mean to sum from $n=1$ to $\infty$? Aug 19, 2018 at 11:46
• If a series converges, then the limit of the general term must be $0$. The converse is not true, as the usual example of the harmonic series shows.
– MasB
Aug 19, 2018 at 11:54

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