Strange result for sum $\sum _{k=1}^{\infty } \frac{\sin (k (k-1))}{k}$

Question

In this sum over $k$

Sum[Sin[k (k - 1)]/k, {k, 1, ∞}]

the result still containes the summation index $k$.

 (* Out 1/2 I (Log[E^-I (E^I - E^(I k))] - Log[E^(-I k) (-E^I + E^(I k))]) *)

What is happening here?

If the sum were divergent, Mathematica would normally return the input.

Sum[1/k, {k, 1, ∞}]

(* Out[148]= $\sum _{k=1}^{\infty } \frac{1}{k}$ *)

Nevertheless plotting the r.h.s. (designated by $f$) as a function of $k$

Plot[2/π f, {k, -2 π + 1, 1.1 + 6 π}, 
 PlotLabel -> "Result of a 'strange sum'", AxesLabel -> {"k", "f(k)"},
  PlotRange -> {{-2 π + 1, 4 π + 1}, All}]

we see that it is discontinuous and in the range from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.

This could be an indication the the sum is divergent delivering values in this range. I have not studied the convergence, but confined myself to the Mathematica question.

Cross reference to the convergence question: https://math.stackexchange.com/q/3466339/198592

Answer 1

This is not a serious bug because of

Sum[Sin[k^2 - k]/k, {k, 1, Infinity}]

$$\sum _{k=1}^{\infty } -\frac{\sin \left(k-k^2\right)}{k} $$

I have strong doubts concerning the existence of a closed-form expression for the sum of the series under consideration.

Addition. Following the documentation to Sum and NSum, I obtain a confirmation of the convergence of the series under consideration:

NSum[Sin[k^2 - k]/k,{k,2,Infinity},AccuracyGoal->1,PrecisionGoal->1, WorkingPrecision -> 20]

0.2