Find the sum to infinity of the series `1^2+2^2+3^2+4^2+oodot`

Explanation

The given series is not an arthimetico-geometric series, because `1^(2),2^(2),3^(2),4^(2)`,… are not in A.P. How ever their successive differences `(2^(2)-1^(2)),(3^(2)-2^(2)),(4^(2)-3^(2)),`….i.e., 3,5,7,… form an A.P. So, the process of finding the sum to infinity of an arthimetico-geometric series will be repeated twice as given below. Let
`S_(oo)=1^(2)+2^(2)x+3^(2)x^(2) +4^(2)x^(3)..oo` (1)
or x`S_(oo)=1^(2)x++2^(2)x^(2)+3^(2)x^(3)+...oo` (2)
Subtracting (2) from (1), we get
`(1-x)S_(oo)=1^(2)+(2^(2)-1^(2))x+(3^(2)-2^(2))x^(2)+(4^(2)-3^(2))x^(3)+...`
or `(1-x)S_(oo)=1+3x+5x^(2)+7x^(3)`+...(3)
his is an arthimatico-geometric series in which a=1,d=2,r=x. Therefore,
`(1-x)S_(oo)=1/(1-x)+(2x)/((1-x)^(2))=(1+x)/((1-x)^(2))`
or `S_(oo)=(1+x)/((1-x)^(3))`