Prove 14n2−1=1(2n+1)(2n−1)=12(2n−1)−12(2n+1)Could you explain the operation in the third step?14n2−1=1(2n+1)(2n−1)=12(2n−1)−12(2n+1)It comes from the sumation∑n=1∞14n2−1I could just copy it to my homework, but I'd like to know how this conversion works. Thanks in advance.

Question

Prove 14n2−1=1(2n+1)(2n−1)=12(2n−1)−12(2n+1)Could you explain the operation in the third step?14n2−1=1(2n+1)(2n−1)=12(2n−1)−12(2n+1)It comes from the sumation∑n=1∞14n2−1I could just copy it to my homework, but I&#039;d like to know how this conversion works. Thanks in advance.

memantangti17 · Accepted Answer

Instead of moving from the second expression to the third, try to move from the third to the second.More generally, this technique is called &quot;decomposing into partial fractions.&quot;In this particular case, you might start by writing:1(2n+1)(2n−1)=A2n+1+B2n−1Now multiply both sides by (2n+1)(2n−1) and solve for A and B.

Prove \(\displaystyle{\frac{{{1}}}{{{4}{n}^{{2}}-{1}}}}={\frac{{{1}}}{{{\left({2}{n}+{1}\right)}{\left({2}{n}-{1}\right)}}}}={\frac{{{1}}}{{{2}{\left({2}{n}-{1}\right)}}}}-{\frac{{{1}}}{{{2}{\left({2}{n}+{1}\right)}}}}\) Could you explain the operation in

Answered question

Answer & Explanation

New Questions in Pre-Algebra