- #1
linda300
- 61
- 3
Hi,
I've been working on this question which asks to show that
[itex]{{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}{d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}}[/itex]
So first taking the n derivatives of the binomial expansions of (x2-1)n
[itex]{{({{x}^{2}}-1)}^{n}}=\sum\limits_{k=0}^{n}{{{(-1)}^{k}}\frac{n!}{k!(n-k)!}{{x}^{2n-2k}}}[/itex]
[itex]\frac{{{d}^{n}}}{d{{x}^{n}}}...=\sum\limits_{k=0}^{n}{{{(-1)}^{k}}\frac{n!}{k!(n-k)!}(2n-2k)(2n-2k-1)...(2n-2k-n+1){{x}^{2n-2k}}}[/itex]
[itex]=n!\sum\limits_{k=0}^{n}{{{(-1)}^{k}}\frac{(2n-2k)!}{k!(n-k)!(n-2k)!}{{x}^{2n-2k}}}[/itex]and comparing it with[itex]{{P}_{n}}(x)=\sum\limits_{m=0}^{M}{{{(-1)}^{m}}\frac{(2n-2m)!}{{{2}^{n}}m!(n-m)!(n-2m)!}{{x}^{n-2m}}},\,\,\,\,M=\frac{n}{2},\frac{n-1}{2}[/itex]
[itex]=\frac{1}{{{2}^{n}}}\sum\limits_{m=0}^{\frac{n}{2}}{{{(-1)}^{m}}\frac{(2n-2m)!}{m!(n-m)!(n-2m)!}{{x}^{n-2m}}}[/itex]
I'm having trouble with the final part,
It's clear that there's a factor of 1/n!2n difference between them but also
the Pn(x) series has m=0...n/2, and also xn , where as the n'th derivative series has k=0...n and x2n.
How can you rewrite one in terms of the other so they both have the same sum limits?
I've tried setting k=2s in the n'th derivative series and a bunch of other similar changes, but non will change the n'th powers of x.
The reason I noticed this was because the last terms of the series arn't the same,
the first series has last term, (-n)! on the bottom, means 1/infinity right?
[itex]n!{{(-1)}^{n}}\frac{0!}{n!0!(-n)!}[/itex]
and the second
[itex]\frac{1}{{{2}^{n}}}{{(-1)}^{\frac{n}{2}}}\frac{n!}{n!(\frac{n}{2})!0!}{{x}^{0}}[/itex]
Have I made a mistake early on or is there a clever way to combine the two series?
Thanks,
Linda
I've been working on this question which asks to show that
[itex]{{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}{d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}}[/itex]
So first taking the n derivatives of the binomial expansions of (x2-1)n
[itex]{{({{x}^{2}}-1)}^{n}}=\sum\limits_{k=0}^{n}{{{(-1)}^{k}}\frac{n!}{k!(n-k)!}{{x}^{2n-2k}}}[/itex]
[itex]\frac{{{d}^{n}}}{d{{x}^{n}}}...=\sum\limits_{k=0}^{n}{{{(-1)}^{k}}\frac{n!}{k!(n-k)!}(2n-2k)(2n-2k-1)...(2n-2k-n+1){{x}^{2n-2k}}}[/itex]
[itex]=n!\sum\limits_{k=0}^{n}{{{(-1)}^{k}}\frac{(2n-2k)!}{k!(n-k)!(n-2k)!}{{x}^{2n-2k}}}[/itex]and comparing it with[itex]{{P}_{n}}(x)=\sum\limits_{m=0}^{M}{{{(-1)}^{m}}\frac{(2n-2m)!}{{{2}^{n}}m!(n-m)!(n-2m)!}{{x}^{n-2m}}},\,\,\,\,M=\frac{n}{2},\frac{n-1}{2}[/itex]
[itex]=\frac{1}{{{2}^{n}}}\sum\limits_{m=0}^{\frac{n}{2}}{{{(-1)}^{m}}\frac{(2n-2m)!}{m!(n-m)!(n-2m)!}{{x}^{n-2m}}}[/itex]
I'm having trouble with the final part,
It's clear that there's a factor of 1/n!2n difference between them but also
the Pn(x) series has m=0...n/2, and also xn , where as the n'th derivative series has k=0...n and x2n.
How can you rewrite one in terms of the other so they both have the same sum limits?
I've tried setting k=2s in the n'th derivative series and a bunch of other similar changes, but non will change the n'th powers of x.
The reason I noticed this was because the last terms of the series arn't the same,
the first series has last term, (-n)! on the bottom, means 1/infinity right?
[itex]n!{{(-1)}^{n}}\frac{0!}{n!0!(-n)!}[/itex]
and the second
[itex]\frac{1}{{{2}^{n}}}{{(-1)}^{\frac{n}{2}}}\frac{n!}{n!(\frac{n}{2})!0!}{{x}^{0}}[/itex]
Have I made a mistake early on or is there a clever way to combine the two series?
Thanks,
Linda
