The sum of the infinite series 1+43+932+1633+…∞ is

This is clearly not an arithmetico-geometric series,since 1, 4, 9, '16, ... are not in AP. However, their successive differences (4 - 1), (9 - 4), (16 - 9) ,... are in AP.LetS∞=1+43+932+1633+…∞13S∞=13+432+933+…∞On subtraction, we get23S∞=1+33+532+733+…∞⇒13⋅23S∞=13+332+533+…∞ multiplying both sides by 13On subtracting the two series, we get49⋅S∞=1+23+232+233+…∞=1+231+13+132+…∞=1+23×11−13=2∴ S∞=2×94=92