Question 780889
The series is:
{{{1/(1 * 3 * 5)+1/(3 * 5 * 7)+1/(5 * 7 * 9)+1/(7 * 9 * 11)}}}+ ...


The n-th term of the series is given by
{{{t[n]=1/((2n-1)(2n+1)(2n+3))}}}
{{{t[n]=(2(2n-1)+(2n+1)+(2n+3))/(2(2n-1)(2n+1)(2n+3))}}}
{{{t[n]=1/((2n+1)(2n+3))+1/(2(2n-1)(2n+3))+1/(2(2n-1)(2n+1))}}}
{{{t[n]=((2n+3)-(2n+1))/(2(2n+1)(2n+3))+((2n+3)-(2n-1))/(8(2n-1)(2n+3))+((2n+1)-(2n-1))/(4(2n-1)(2n+1))}}}
{{{t[n]=1/(2(2n+1))-1/(2(2n+3))+1/(8(2n-1))-1/(8(2n+3))+1/(4(2n-1))-1/(4(2n+1))}}}
{{{t[n]=1/(4(2n+1))-5/(8(2n+3))+3/(8(2n-1))}}}


The sum of the given series up to infinity is given by
{{{S[infinity]=sum(( 1/(4(2n+1))-5/(8(2n+3))+3/(8(2n-1)) ), n=1, infinity )}}}


I have given you the clue. From here see if you can use your brain and proceed...