Hi there!
This is my problem
Calculate $\displaystyle \sum_{k=1}^n \frac{1}{(2k-1)(2k+1)}$ And then prove it, with help of induction?
Can any one of you help me with this problem?
if you get to the expression it should be equal to, can you do the induction?
this is a telescoping sum. meaning, you can write it in a different form that when you expand it, most of the terms will cancel out, leaving you with what the sum actually is.
note that $\displaystyle \frac 1{(2k - 1)(2k + 1)} = \frac 12 \bigg( \frac 1{2k - 1} - \frac 1{2k + 1} \bigg)$ (if you can't expand like that using algebra, just use partial fractions)
so that $\displaystyle \sum_{k = 1}^n \frac 1{(2k - 1)(2k + 1)} = \frac 12 \sum_{k = 1}^n \bigg( \frac 1{2k - 1} - \frac 1{2k + 1} \bigg)$
now expand the sum on the right to see what cancels out. that will get you your formula to do the induction with