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 2022-04-02 Answered
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.
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Answers (1)

memantangti17
Answered 2022-04-03 Author has 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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