Text and Tests 5

analyse this

Chapter 9 section 9C of Text and Tests 5 which examines the Binomial Series. Is there an error where the book is discussing the General Term of a binomial expression using the Maclarin Series:o I just don't get it!!!

cocoa

quick answer, yes, i remember having to fix it in my book, I'll go look at the book and tell you exactly in a few mins...

ok, in my book it's in section 9E but anyway...

for me, it says Ur=n(n-1)(n-2)...(n-r) x^r-1 / (r-1)!
replace (n-r) with (n-r+2)
the formula for Ur+1 is now correct.
you also have to add in the 2 in the example.
and that's it.

genericgoon

Id prefer if we get another easy maclurain this year again.

analyse this

They are not particularly that hard! There seems to an awful stigma against them!

Lucas10101

Maclaurin Series is very easy and just a heap of memory work. I Know of an exponential part for (a) which you just derived it and got 15 Marks? I ope there isn't a difficult C part on Maximum/Minimum problems.

What's this with the change in Text and Tests 5??? I learned the book method off by heart and don't understand the proposed error that's being offered?

Clarification please?

analyse this

The book states the general term of the binomial expansion (1+X)^n is:

Un= n(n-1)(n-2)....(n-r)/(r-1)! by x^r-1

However, if you inspect the terms you will see that the general term doesn't work.

The actual general term is :

Un= n(n-1)(n-2)...(n-r+2)/(r-1)! by x^r-1

analyse this

I ope there isn't a difficult C part on Maximum/Minimum problems

I love them!!!:p

analyse this

ok, in my book it's in section 9E but anyway...

Apologies:o

Lucas10101

could you write out the whole binomial expansion for the Ratio Test for this please? I really can't see the mistake and want to see it written out fully as I can't do it.

cocoa

Lucas10101 wrote:

could you write out the whole binomial expansion for the Ratio Test for this please? I really can't see the mistake and want to see it written out fully as I can't do it.

do you have the book? the books explanation is fine, it's just wrong in that one aspect. Try (n+1) in the place of n for their version, and then our verion, you'll see the mistake. What the did was replace -n with -n + 1 which is incorrect...

kisaragi

Does this mean that the term Ur+1 should be..

n(n-1)(n-2)...(n-r+3) x^r
r!

I'm totally lost too...

rjt

Ah, the joys of doing Group Theory

kisaragi

Lol my maths teacher never told us about this mistake... infact I'm pretty sure he just glanced over it with us and told us it wouldn't come up...

MathsManiac

Lucas10101 wrote:

could you write out the whole binomial expansion for the Ratio Test for this please? I really can't see the mistake and want to see it written out fully as I can't do it.

I don't have the book you're talking about, but I can do that for you.

Since you didn't specify, I'm assuming you're expanding (1+x)^m, which is how it's given in the syllabus. And I'll use n for the indexing letter, as is usually done with MacLaurin expansions. (You can change the m to n and the n to r if you want, but then you can't call the general term Un; you'll have to call it Ur.)
I'm assuming that you like to write your general term in such a way that the terms start with n=1. (Some people prefer to start at n=0, which is fine too, but you get a different form).
On this assumption, the general term is Un= m(m-1)(m-2)...(m-n+2) / (n-1)! times x^(n-1).

Now, the next bit of answering your question depends on what form you like to use for the ratio test. Let's suppose you do it using U(n+1)/U(n). So you want to know what's U(n+1). It is:
U(n+1)= m(m-1)(m-2)...(m-n+1) / (n)! times x^n.

Then when you get U(n+1)/U(n) you're left with (m-n+1)/n times x,
and I assume you can finish from here to show convergence for |x|<1.

Lucas10101

I still don't understand. Not joking here.

I would write:

Ur = n(n-1)(n-2).......(n-r).x^r-1/(r-1)!

Ur+1/Ur => n(n-1)(n-2).....(n-r+1).x^r/r! * (r-1)/n(n-1)(n-2)...(n-r) . 1/x^r-1

n-r+1.x.(r-1)!/r(r-1)!

n-r+1.x/r

Taking the limit makes it Convergent,

This looks messy but write it down and change it tonight or something and copy the new one and paste it in....I'll be back here tomorrow morning to see it and remember it. Thanks

MathsManiac

Lucas10101 wrote:

Ur = n(n-1)(n-2).......(n-r).x^r-1/(r-1)!

That first sequence of brackets should stop at (n-r+2), not at (n-r).

Lucas10101 wrote:

Ur+1/Ur => n(n-1)(n-2).....(n-r+1).x^r/r! * (r-1)/n(n-1)(n-2)...(n-r) . 1/x^r-1

You've (strangely?) managed to get your Ur+1 correct even though your Ur was wrong. [Also, you missed a factorial sign after (r-1) in your inverted Ur, but I assume that's just a typo.]

Lucas10101 wrote:

n-r+1.x.(r-1)!/r(r-1)!

n-r+1.x/r

These lines are also correct on the basis of the correct Ur, (ignoring bracket issues!) but they wouldn't be correct on the basis of your incorrect Ur (because the cancellation wouldn't work out right).

Lucas10101 wrote:

Taking the limit makes it Convergent,

Since the mod of this converges to mod x, I'm assuming you meant to say convergent for |x|<1, in which case you're okay. You could be penalised for asserting that it's convergent for all x, (because it's not,) and saying that it's convergent without specifying would usually be interpreted as an assertion that it's convergent for all x.

Gangsta

Lads the general term in the book is also in the log tables, how could it be wrong then?

cocoa

Gangsta wrote:

Lads the general term in the book is also in the log tables, how could it be wrong then?

umm, which page? I've never seen any maclaurin series in the log tables...

Gangsta

cocoa wrote:

umm, which page? I've never seen any maclaurin series in the log tables...

p35, doesn't really matter now I suppose. They took all the fun out of it by giving it to you!

cocoa

Gangsta wrote:

p35, doesn't really matter now I suppose. They took all the fun out of it by giving it to you!

that's the formula for nCr, not what we were talking about...

Gangsta

same thing if you examine it. All you do is add x^r to get the general term.

cocoa

Gangsta wrote:

same thing if you examine it. All you do is add x^r to get the general term.

you're missing the point, we never disputed the general term (Ur+1) given in the book. We're disputing the Ur given in the book.

Gangsta

My point was don't bother with the book, it's in the log tables.

cocoa

Gangsta wrote:

My point was don't bother with the book, it's in the log tables.

you show me where the expansion of the maclaurin series for (1+x)^m is in the log tables and i will agree.

JC 2K3

Guys, the maths exam is over.

cnolan

does any 1 know where to get an old version of text and texts 5..... for the further probability option Q9 in higher maths ...........
cnolan

cnolan

cnolan wrote: »

does any 1 know where to get an old version of text and texts 5..... for the further probability option Q9 in higher maths ...........
cnolan

Best thing to do is to ask your teacher; they're usually more than willing to give it out, they always like to see students wanting to have a look at the other options.