Question

If `S_(1), S_(2), S_(3),...,S_(n)` are the sums of infinite geometric series, whose first terms are 1, 2, 3,.., n and whose common rations are `(1)/(2), (1)/(3), (1)/(4),..., (1)/(n+1)` respectively, then find the values of `S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2)`.

Answer 1

By the summation of an infinite geometric series, we get
`S_(1)=1/((1-1/2))=2, S_(2)=2/((1-1/3))=3, S_(3)=3/((1-1/4))=4, ...,`
`S_(p)=p/((1-1/(p+1)))=(p+1)`.
`:. S_(1)+S_(2)+S_(3)+...+S_(p)=[2+3+4+...+(p+1)]`
`=p/2.{2+(p+1)}=1/2 p(p+3)`.
Hence, `S_(1)+S_(2)+S_(3)+...+S_(p)=1/2 p(p+3)`.