Question

1 -1/8 + 1.3/8.16 - 1.3.5/8.16.24 ...........
ans. 2/5^1/2

Answer 1

Calculate.
Please show your work
1 - 1/8 + 1.3/8.16 - 1.3.5/8.16.24 ...........
ans. 2/5^1/2

1 - 1/8 + 1.3/8.16 - 1.3.5/8.16.24 ... =& 1 -& (1/2)(1/4) + (1.3/2!.2.2)(1/4)^2- (1.3.5/3!.2.2.2)(1/4)^3 + ... ,
which is of the form
1 - nx + [n.(n+1)/2!]x^2 - [n.(n+1)(n+2)/3!]x^3 ... ,
where n=1/2 and x=1/4, and
1 - nx + [n.(n+1)/2!]x^2 - [n.(n+1)(n+2)/3!]x^3 ... = (1+x)^(-n).
Therefore, 1 - 1/8 + 1.3/8.16 - 1.3.5/8.16.24 ... = (1+1/4)^(-1/2) = (5/2)^(-1/2) = (2/5)^(1/2).