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Binomial expansion

by Appleton
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Appleton
#1
Jun29-14, 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 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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Simon Bridge
#2
Jun29-14, 09:37 AM
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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?
That's what it looks like to me - the author is making a step in the calculation explicit.

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?
micromass
#3
Jun29-14, 09:40 AM
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Quote Quote by Appleton View Post
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.
You likely didn't misunderstand anything, the book just has been incomplete. The book should have mentioned that we define ##{}^nC_m = 0## for ##m< 0## and ##m>n##.

AlephZero
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Jun29-14, 09:52 AM
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Binomial expansion

Quote Quote by Appleton View Post
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?
Yes.

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?
I think the book is a bit careless there. ##{}^{n}C_{k}## is normally only defined for ##0 <= k <= n##. But the only "sensible" defintiion when ##k < 0## or ##k > n## is zero. If you define ##{}^{n}C_{k}## as the number of ways to choose objects from a set, there are no ways to choose more than n different objects from a set of n, and you can't choose a negative number of objects. If you define it using Pascal's triangle, any numbers "outside" the triangle need to be 0 to make the formulas work properly.
Simon Bridge
#5
Jun29-14, 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?
Appleton
#6
Jun29-14, 12:56 PM
P: 40
Thank you so much for clarifying that for me.
Alicelewis11
#7
Jul17-14, 11:30 AM
P: 5
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...
Simon Bridge
#8
Jul18-14, 12:17 AM
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Quote Quote by Alicelewis11 View Post
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?
This question has already been asked and answered - see post #3.


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