But , it came in GATE . So, better to do. right??

I think syllabus of calculus was different that time. Please check some questions of 1993 below, It involved Fourier series, double integral etc.

With stirling formula it came 3. Then why radius of converge inverse the value??

please check it : https://en.wikipedia.org/wiki/Radius_of_convergence#Theoretical_radius

and u initialized

m=0, but it can bem=1 too. right??

yes.