find the sum of the series
1-\frac{1.3}{2.4}+\frac{1.3.5.7}{2.4.6.8}-\frac{1.3.5.7.9.11}{2.4.6.8.10.12}.......\infty
forgive teh joker if it has been asked befor
btw its not a doubt[6]
1giving the hint will mean giving the soln [4]
anyways it involves bino + complex
(a complex no)some -ve fraction
now use some bino and then some complex[1]
1( 1 + x )n = 1 + nx + n ( n - 1 ) x21 . 2 + n ( n - 1 ) ( n - 2 ) x31 . 2 . 3 + .......
where n is any rational index .
comparing with the given series , nx = - 3 / 8 and
n ( n - 1 ) x2 / 2 = nx ( nx - x ) / 2 = 35 / 128
solving for x and n , I got x = 13 / 12 and n = - 9 / 26.
So i think the answer is ----- ( 1 + 13 / 12 ) - 9 / 26
I checked the answer using a calculator , and its giving correct results
1ok i m giving teh ans
expand (1+i)-1/2 using bino
and then equale real part of (1+i)-1/2 with that of its expansion
the real part of the expansion is teh given series
1but my answer is also matching with your series --- can you spot the flaw ?