# Some of the values of $f(x)$ are different from those derived by Table

It is known that there are the following functions:

f[x_] := x - 1/2 + 1/Pi Sum[Sin[2 Pi k x]/k, {k, 1, Infinity}]


Tests on some of the values taken.

f[0] // Chop
f[0.5] // Chop
f[1] // Chop
f[1.5] // Chop
f[2] // Chop


-(1/2), 0, 1/2, 1., 3/2

But with Table, the result will be different.

Table[f[x], {x, 0, 2, 0.5}] // Chop


{-0.5, 0, 0, 1., 1.}

It is clear that the values of the function at $$x=1$$ and $$x=2$$ cannot correspond, why is that?

• It comes from this Sum[Sin[2 Pi k x]/k, {k, 1, Infinity}], here is screen shot to illustrate !Mathematica graphics Seems to have caused some issue here. Need more investigation to find exactly why. But as a general rule, Try to avoid non exact numbers when doing exact calculations. Aug 31, 2023 at 13:25
• The results in the picture shocked me, first time I've had this problem @Nasser. Aug 31, 2023 at 13:30
• Yes, seems to be some numerical issue in the sum you have. Could be a bug. But need more looking into it to know exactly why it happened. (but as I said, to be safe, avoid non-exact numbers in exact calculations). Aug 31, 2023 at 13:32

Your function is discontinuous on the integers, and so it matters exactly where you are evaluating it. Numerical is not equal to analytical evaluation for discontinuous functions.

f[x_] = x - 1/2 + 1/π Sum[Sin[2 π k x]/k, {k, 1, ∞}]
(*    -1/2 + x + (I (Log[1 - E^(2 I π x)] - Log[E^(-2 I π x) (-1 + E^(2 I π x))]))/(2 π)    *)

Plot[f[x], {x, 0, 10}]


Show the discontinuity at $$x=2$$ for example:

Limit[f[x], x -> 2, Direction -> "FromBelow"]
(*    1    *)

Limit[f[x], x -> 2, Direction -> "FromAbove"]
(*    2    *)


So, numerically, you may get 1, 2, or their average 1.5 depending on details.