Math Is Fun Forum

nevinsmith

Member

Registered: 2010-05-19

Posts: 39

Help with binomial expansion!

so the question is expand 1 / (x^2+3)

so i take 1st term as x^2, 2nd term as 3 and power as -1

using binomial series i got-

1/3-(x^2)/9+(x^4)/27-(x^6)/81.....            true for all values of x???

but the answer says-

3-9x^2+27x^4-81x^6....                        true for -sqrt3>x>sqrt3

??? why does that happen?
is it to do with requirements of convergence or am i just wrong?

so how do i calculate the valid values of x for convergence?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Help with binomial expansion!

hi nevinsmith

stay on-line answer coming up

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Help with binomial expansion!

hi nevinsmith

You have

You can only use the general form of the binomial theorem for n = -1 and this form also requires a 1 not a 3 in the bracket.

So do this:

Now you can proceed with the expansion.

Post again if you need more help

Bob

Last edited by Bob (2011-04-12 20:07:49)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nevinsmith

Member

Registered: 2010-05-19

Posts: 39

Re: Help with binomial expansion!

hi bob bundy, thanks for helping

so i expand 1/3(1/3x2+1)^-1

i still get the terms in fractions with convergence of every value of x

1/3-(x^2)/9+(x^4)/27-(x^6)/81.....           

can you please explain how the answer is different?

which is              3-9x^2+27x^4-81x^6....                        true for -sqrt3>x>sqrt3

how did the terms not have fractions?

thanks for your reply

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Help with binomial expansion!

hi again,

n = -1    'x' = x^2 / 3

I think I've sorted it now :

Sorry.  I'm not getting a fractionless result.  I've got to go out now but I'll look again in about 3 hours time (1pm BST)

Bob

Last edited by Bob (2011-04-12 20:51:59)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help with binomial expansion!

Hi nevinsmith;

Did you try a value for x in that supposed series without denominators? You know, between -√3 and √3. And then try that value in 1 / (x^2 + 3)?

It certainly is not the series expansion for 1 / (x^2 + 3).

Actually:

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Help with binomial expansion!

hi nevensmith

I stand by my answer:

Is the answer you have from a book.  I would suspect a printing error.

The test is "Does either formula actually work"?

Here's my figures

Pick a value for x in the range let's say x = 0.5

value of original expression = 0.307692307692308

Target answer:  the values for the first five terms  3 - 2.25 + 1.6875 -1.265625 +0.94921875 = 2.12109375

not even close!

My answer = 0.333333333333333  -0.0277777777777778 + 0.00231481481481481 -0.000192901234567901 + 0.0000160751028806584 = 0.307693544238683

That's close enough for me.  Take more terms if you want to get closer

this is the ultimate test.  Does the expansion actually work?

The range is found by having -1 < (x^2)/3 < 1 and re-arranging to get -root 3 < x < root 3

Hope that helps,

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Help with binomial expansion!

Hi all,

Looking at the answer which the source says,
there's probably a mistake with the question.
It may be

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Help with binomial expansion!

hi gAr

You may be right but then

true for -sqrt3>x>sqrt3

isn't right.

edit: In fact, now I look at it again, it isn't right anyway.  The signs are back to front!

Bob

Last edited by Bob (2011-04-13 00:59:26)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Help with binomial expansion!

Hi Bob,

Oh, didn't check that condition when posting.

edit: In fact, now I look at it again, it isn't right anyway.  The signs are back to front!

Yes, I observed it as well!

Looks like multiple mistakes then.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help with binomial expansion!

Hi nevinsmith, Bob, gAr, bobbym;

I stand by my answer:

I did not explain myself right. I was not questioning your answer. That is what I got, so it has to be right. Why would I get a wrong answer? What I do not like is  3-9x^2+27x^4-81x^6.... as the answer . As I showed in post #6 that is not the series for 1 / (x^2+3)

Hi nevinsmith;

Math results have to withstand the test of numbers. Numerics is part of math too. Whenever you get a series representation for some expression always check to see that it is at least close. Then produce a few more terms of your series and convince yourself that it is converging to the correct answer. There are more precise means but this type of testing will weed out most of the errors quickly.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Help with binomial expansion!

Good morning bobbym,

I did not explain myself right. I was not questioning your answer.

I wasn't upset, thinking you were criticising me.  I had already made up that post in my head while I was out of the house.  Oh yes, and I knew I was right too!  (confidence = 90%)

That is what I got, so it has to be right. Why would I get a wrong answer?

Well now I'm totally confident! (confidence = 99.99%)   Together, we must be on top of the Pyramid of Correctness.  You can shapen razor blades just by holding them close. 

Best wishes. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help with binomial expansion!

Yes! Pyramid power!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

nevinsmith

Member

Registered: 2010-05-19

Posts: 39

Re: Help with binomial expansion!

yea i would think there is a print error

the origional problem was probably what gAr said it was

the book is pretty darn useless at explaining stuff
thanks for the help guys!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help with binomial expansion!

Hi nevinsmith;

What is the name of that book?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.