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

jisu61hbke

jisu61hbke

Answered question

2022-04-02

Prove 14n21=1(2n+1)(2n1)=12(2n1)12(2n+1)
Could you explain the operation in the third step?
14n21=1(2n+1)(2n1)=12(2n1)12(2n+1)
It comes from the sumation
n=114n21
I could just copy it to my homework, but I'd like to know how this conversion works. Thanks in advance.

Answer & Explanation

memantangti17

memantangti17

Beginner2022-04-03Added 13 answers

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 "decomposing into partial fractions."
In this particular case, you might start by writing:
1(2n+1)(2n1)=A2n+1+B2n1
Now multiply both sides by (2n+1)(2n1) and solve for A and B.

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