taking the limit of an infinite series

[dts5044]

what is the absolutely correct, most rigorous way to prove:

lim [summation of n =1 to infinity] t[sup:286xypl0]2n[/sup:286xypl0]/n[sup:286xypl0]2[/sup:286xypl0]
t---> 1[sup:286xypl0]-[/sup:286xypl0]

equals [summation of n =1 to infinity] 1/n[sup:286xypl0]2[/sup:286xypl0]

???

 

[GeneralSynopsis]

what is the absolutely correct, most rigorous way to prove:

lim [summation of n =1 to infinity] t[sup:17woopoy]2n[/sup:17woopoy]/n[sup:17woopoy]2[/sup:17woopoy]
t---> 1[sup:17woopoy]-[/sup:17woopoy]

equals [summation of n =1 to infinity] 1/n[sup:17woopoy]2[/sup:17woopoy]

???

Application of Abel's theorem