Binomial series expansion

  • Thread starter sooyong94
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  • #26
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Yes, but you should simplify the signs, as you've been shown.

the coefficient of ##x^{400}## is
Does it sound right to you?
##\frac{(2)(3)(4)...(399}{400!} (4)^{400}##
 
  • #27
haruspex
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the coefficient of ##x^{400}## is
Does it sound right to you?
##\frac{(2)(3)(4)...(399)}{400!} (4)^{400}##
No, you've made a mistake in the arithmetic (the 399).
Having fixed that, there's another simplification to do (which you did much earlier in the thread, post #9).
 
  • #28
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No, you've made a mistake in the arithmetic (the 399).
Having fixed that, there's another simplification to do (which you did much earlier in the thread, post #9).

Why? :confused:
 
  • #30
Simon Bridge
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I have pretty much the same notes for you:
So... that would look something like this...
the coefficient of ##x^{400}## is
##\frac{(-2)(-3)(-4)...(-400-2+1)}{400!} (-4)^{400}##
Therefore:
the coefficient of ##x^{400}## is
Does it sound right to you?
##\frac{(2)(3)(4)...(399}{400!} (4)^{400}##
... ##-400-2+1\neq -399## check arithmetic.

You have to remember to use what you learned in previous posts when you do future ones.
These lessons all fit together.

i.e. the numerator in the last calculation can be written in a shorter way remember?
This leads to a simpler way to write the whole fraction part.

Once you've done that - you need to compare what you got with what you have to show way back in post #1. If you go reread that but you should see the pattern already.
 
  • #31
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I have pretty much the same notes for you:
Therefore:
... ##-400-2+1\neq -399## check arithmetic.

You have to remember to use what you learned in previous posts when you do future ones.
These lessons all fit together.

i.e. the numerator in the last calculation can be written in a shorter way remember?
This leads to a simpler way to write the whole fraction part.

Once you've done that - you need to compare what you got with what you have to show way back in post #1. If you go reread that but you should see the pattern already.

-401.... :oops: Ok.. I get it now. :P
 
  • #32
Simon Bridge
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Great - so now all you have to do is put it all together!
Remember what you are trying to show?
 
  • #33
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Great - so now all you have to do is put it all together!
Remember what you are trying to show?
##\frac{(2)(3)(4)...(401)}{400!} (4)^{400}##

##=\frac{(1)(2)(3)(4)...(401)}{400!} 4^{400}##
##=\frac{(401!)}{400!} 4^{400}##
##=\frac{(401)(400!)}{400!} 4^{400}##
##=401 (4)^{400}##

##a=401, k=400##
 
  • #34
haruspex
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##\frac{(2)(3)(4)...(401)}{400!} (4)^{400}##

##=\frac{(1)(2)(3)(4)...(401)}{400!} 4^{400}##
##=\frac{(401!)}{400!} 4^{400}##
##=\frac{(401)(400!)}{400!} 4^{400}##
##=401 (4)^{400}##

##a=401, k=400##
That's it.
 
  • #35
Simon Bridge
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Beautiful - pour yourself a drink!
 

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