A Suspicious Series

Does the series

\displaystyle \sum_{k=1}^\infty \frac{\sin k}{k}

converge?

At first, you may be reminded of the harmonic series that diverges, because of the divisor k following the same progression, and may conclude that this suspicious series diverges because its terms do not go to zero fast enough. But we need to investigate how the \sin k part behaves.

The sine part changes sign, and somewhat smoothly, despite k being an integer value rather than some nice continuous quantity varying between 0 and 2 \pi. If we plot the first few terms of the series (not its partial sum, its terms), we get the following graph:

series-integer-sine

So the sine part changes sign, but not wildly. There seems to be an equally number of positive and negative values, they kind of alternate, and their magnitudes reduce rapidly. So maybe the sum does’t diverge afterall because you add some, subtract some, more or less alternating, and with ever smaller numbers. But we may be concluding too rapidly: maybe there’s an uneven number of additions and subtractions and it still diverges. Let’s see how it behaves numerically. Computing the partial sum

\displaystyle s_n = \sum_{k=1}^n \frac{\sin k}{k},

we get the following graph:

series-sums-all

The abscissa gives the number of terms in the sum. If we look at it more closely,

series-sums-detail

we see that it oscillates around a precise value. Let’s have a look at it numerically rather than graphically. Using Mathematica using 50-digits precision floating point numbers, I got the following table:

10^k
terms
Sum
1 1.124518813379952827526344957264684482544…
2 1.060428938401062128124909620302077101280…
3 1.070694154319583908203085543442136462838…
4 1.070868194833174420081558343499182371133…
5 1.070805652123434076823901713055378434225…
6 1.070795294441892344626876203420384462086…
7 1.070796430859646813540329599847671757781…

The series converges to a constant, \approx 1.07079. This constant, turns out, and after quite a bit of calculus, is exactly \displaystyle \frac{\pi-1}{2}=1.0707963267948966192313216916\ldots.

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