I am puzzled by the following example of the application of binomial expansion from Bostock and Chandler's book Pure Mathematics:
If n is a positive integer find the coefficient of xr in the expansion of (1+x)(1-x)n as a series of ascending powers of x.
[itex](1+x)(1-x)^{n} \equiv (1-x)^{n} + x(1-x)^{n} [/itex]
[itex]\equiv\sum^{n}_{r=0} { }^{n}C_{r}(-x)^{r} + x\sum^{n}_{r=0} { }^{n}C_{r}(-x)^{r}[/itex]
[itex]\equiv\sum^{n}_{r=0} { }^{n}C_{r}(-1)^{r} x^{r}+ \sum^{n}_{r=0} { }^{n}C_{r}(-1)^{r}x^{r+1}[/itex]
[itex]\equiv [1-{ }^{n}C_{1}x+{ }^{n}C_{2}x^{2}...+{ }^{n}C_{r-1}(-1)^{r-1} x^{r-1}+{ }^{n}C_{r}(-1)^{r} x^{r}+...+(-1)^{n}x^{n}][/itex]
[itex]+[x-{ }^{n}C_{1}x^{2}+...+{ }^{n}C_{r-1}(-1)^{r-1} x^{r}+{ }^{n}C_{r}(-1)^{r} x^{r+1}+...+(-1)^{n}x^{n+1}][/itex]
[itex]\equiv\sum^{n}_{r=0} [{ }^{n}C_{r}(-1)^{r} + { }^{n}C_{r-1}(-1)^{r-1}]x^{r}[/itex]
The 4th and 5th line seemed a peculiar way of writing it. Were they just trying to demonstrate how the second series is always one power of x ahead?
The last expression seems to require a definition of [itex]{ }^{n}C_{-1}[/itex] which hasn't been defined in the book so I'm guessing I have misunderstood something. Could someone please explain this for me?
Apologies for any typos, I'm using a mobile. Very fiddley.
If n is a positive integer find the coefficient of xr in the expansion of (1+x)(1-x)n as a series of ascending powers of x.
[itex](1+x)(1-x)^{n} \equiv (1-x)^{n} + x(1-x)^{n} [/itex]
[itex]\equiv\sum^{n}_{r=0} { }^{n}C_{r}(-x)^{r} + x\sum^{n}_{r=0} { }^{n}C_{r}(-x)^{r}[/itex]
[itex]\equiv\sum^{n}_{r=0} { }^{n}C_{r}(-1)^{r} x^{r}+ \sum^{n}_{r=0} { }^{n}C_{r}(-1)^{r}x^{r+1}[/itex]
[itex]\equiv [1-{ }^{n}C_{1}x+{ }^{n}C_{2}x^{2}...+{ }^{n}C_{r-1}(-1)^{r-1} x^{r-1}+{ }^{n}C_{r}(-1)^{r} x^{r}+...+(-1)^{n}x^{n}][/itex]
[itex]+[x-{ }^{n}C_{1}x^{2}+...+{ }^{n}C_{r-1}(-1)^{r-1} x^{r}+{ }^{n}C_{r}(-1)^{r} x^{r+1}+...+(-1)^{n}x^{n+1}][/itex]
[itex]\equiv\sum^{n}_{r=0} [{ }^{n}C_{r}(-1)^{r} + { }^{n}C_{r-1}(-1)^{r-1}]x^{r}[/itex]
The 4th and 5th line seemed a peculiar way of writing it. Were they just trying to demonstrate how the second series is always one power of x ahead?
The last expression seems to require a definition of [itex]{ }^{n}C_{-1}[/itex] which hasn't been defined in the book so I'm guessing I have misunderstood something. Could someone please explain this for me?
Apologies for any typos, I'm using a mobile. Very fiddley.
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