Power series, partial sums.  

The following defines partial sums for power series and plots the partial sum.  You can/should change the coefficients as appropriate aa[k_]:=  ... 

In  "partialsum[ _ , _ ]" for the plot, the second variable is the max power of x in the sum (the number of terms will be one more)
Generally speaking, you should change the plot domain {x, __ , __ } to reflect the radius of convergence.   

PlotRange sets the vertical scale, which you may want to adjust depending on the particular series.

aa[k_]:=(k!)^2/(2k)!;partialsum[x_,n_]:=Sum[aa[k]x^k,{k,0,n}];Plot[partialsum[x,20],{x,-6,6},PlotRange->{-5,20}]