In / Out numbers are kernel specific and not preserved in a saved notebook.

This error is self-explanatory, take a look at the domain of fun

InterpolatingFunction::dmvali: The integration endpoint 0 in dimension 1 lies outside the range of data in the interpolating function. Extrapolation will be used.

This one is also clear because nx is not defined.

NIntegrate::inumr: The integrand Cos[24.3054 nx] <<1>> has evaluated to non-numerical values for all sampling points in the region with boundaries {{0.30383,0.30837}}.

I still do not understand how to 'parameterize the extraction of n terms', Rohit - this is something I do not know how to do and would really appreciate if you could show me what you mean.

Thank you,

Crossposted here.

One more point - the third period seems to be fairly accurate. Any chance we can get the first period (or the entire thing!) to be more accurate?

1) Plot can plot several functions in the same plot

Plot[{fun[x], recon[x]}, {x, x1, x2}]

Use FirstPosition to locate the position of the first Cos and Sin

firstCos = FirstPosition[recon[x], _Cos] // First
firstSin = FirstPosition[recon[x], _Sin] // First
fiveCos = recon[x][[firstCos ;; firstCos + 4]]
fiveSin = recon[x][[firstSin ;; firstSin + 4]]   

recon5[x_] = recon[x][[;; firstCos - 1]] + fiveCos + fiveSin
Plot[{fun[x], recon5[x]}, {x, x1, x2}]

2)

NIntegrate[fun[x]*Cos[x], {x, x1, x2}] (* Change limits as needed *)
(* 0.000955089 *)

3) For the manipulate, parameterize the extraction of n terms as was done for 5 terms above. Try it and if you get stuck, post what you tried.