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Binomial expansion 
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#1
Jun2914, 09:11 AM

P: 40

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 x^{r} in the expansion of (1+x)(1x)^{n} as a series of ascending powers of x. [itex](1+x)(1x)^{n} \equiv (1x)^{n} + x(1x)^{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_{r1}(1)^{r1} x^{r1}+{ }^{n}C_{r}(1)^{r} x^{r}+...+(1)^{n}x^{n}][/itex] [itex]+[x{ }^{n}C_{1}x^{2}+...+{ }^{n}C_{r1}(1)^{r1} 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_{r1}(1)^{r1}]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. 


#2
Jun2914, 09:37 AM

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Do you see how the last line is derived from the one before it? Notes: ...everything from the third "equivalence" sign to (but not including) the fourth one is all one line of calculation. Do Bostock and Chandler number their working, their equations? 


#3
Jun2914, 09:40 AM

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#4
Jun2914, 09:52 AM

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Binomial expansion



#5
Jun2914, 10:13 AM

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The definition being used should be evident by following the derivation though... looking at the coefficient of x^0, probably why the authors felt they could be a bit sloppy there?



#6
Jun2914, 12:56 PM

P: 40

Thank you so much for clarifying that for me.



#7
Jul1714, 11:30 AM

P: 14

The last articulation appears to oblige a meaning of nc−1 which hasn't been characterized in the book so I'm speculating I have misconstrued something. Would someone be able to please clarify this for me?
Expressions of remorse for any typos, I'm utilizing a versatile. Exceptionally fiddle... 


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