I believe that it converges. And I am confused about why $$\int ^{\infty} _{-\infty}e^{-ikx}dk=0$$ if $x \neq 0$.
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https://en.wikipedia.org/wiki/Dirac_delta_function#History – Andrei Oct 29 '22 at 12:55
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Did you mean $\int ^{\infty} _{-\infty}e^{ikx}dx$? – neo Oct 29 '22 at 12:58
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@neo I think he really meant what he wrote (since he specified "if $x\ne0$"). But this question is indeed equivalent to the duplicate you propose. – Anne Bauval Oct 29 '22 at 13:22
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No, this integral as it is does not converge. The function $S(a,b)=\int_a^be^{-ikx}dk$ is for $k\ne 0$ non-constant periodic in $a,b$ and does thus not have a limit for $(a,b)\to(-\infty,\infty)$ – Lutz Lehmann Oct 30 '22 at 10:46