Ask WolframAlpha to evaluate
(21-n)^(it)*(21-n)^Sum[i^m*pi^m/m!,{m,1,n}]
Click to see the Mathematica plain text that you can scrape
(21-n)^(-1+I t-Gamma[1+n,I Pi]/Gamma[1+n])
Copy and paste that to ask WolframAlpha to evaluate
Plot[Sum[(21-n)^(-1+I t-Gamma[1+n,I Pi]/Gamma[1+n]),{n,2,19}],{t,0,40}]
and you get
Your desired plot
I hope I have correctly understood what you are trying to calculate.
