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# Maclauren's Series ln(1+x) Watch

1. I am looking at Stroud Engineering Mathematics Programme 14 Further Question 2, which says

"Apply Maclaurin's series to establish a series for ln(1+x). If 1+x=b/a, show that (b^2-a^2)/2ab=x-x^/2+x^3/2-... [It does say x^3/2 and I wonder if that's a typo].
Hence show that if b is nearly equal to a, then (b^2-a^2)/2ab exceeds ln(b/a) by approximately (b-a)^3/6a^3"

I can get ln(1+x) = x-x^2/2+x^3/3-... but I am stumped as where to go next.

Can anyone point me in the right direction?

Thanks
Ant
2. (Original post by antlee)
I am looking at Stroud Engineering Mathematics Programme 14 Further Question 2, which says

"Apply Maclaurin's series to establish a series for ln(1+x). If 1+x=b/a, show that (b^2-a^2)/2ab=x-x^/2+x^3/2-... [It does say x^3/2 and I wonder if that's a typo].

As long as you realise the "/2" is not part of the exponent, then no, it's not a typo.

Divide out the left hand side of your equality, and you'll see how it's constructed from b/a.

Note: This bit has nothing to do with the log part, yet.
3. Yes, yes, yes...

I get that bit now

(b^2-a^2)/2ab

= 1/2(b^2/ab-a^2/ab)

= 1/2(b/a - a/b)

= 1/2((1+x)-(1+x)^-1)

= 1/2 (1+x - (1-x+x^2-x^3+....))

= 1/2 (2x - x^2 + x^3 - ...)

= x - x^2/2 + x^3/2 - ....

So not a typo...

Thanks very much.

I may come back for further help on the last bit but very happy at the moment.

Ant
4. Ghostwalker

So now I get as b is close to a that

(b^2-a^2)/2ab-ln(b/a)

= (x-x/2+x^3/2-...)-(x-x/2+x/3-...)

= x^3(1/2-1/3) = x^3/6

= (b/a-1)^3/6

=((b-a)/a)^3/6

= (b-a)^3/6a^3

Ta dah!

Thanks for the hint I'm very pleased as I was then able to finish it off myself.

Cheers
Ant
5. (Original post by antlee)
Ghostwalker

So now I get as b is close to a that

(b^2-a^2)/2ab-ln(b/a)

= (x-x/2+x^3/2-...)-(x-x/2+x/3-...)

= x^3(1/2-1/3) = x^3/6
You want to be careful here, as it doesn't equal x^3(1/2-1/3).

We can say ignoring higher order terms

etc.

Thanks for the hint I'm very pleased as I was then able to finish it off myself.

Cheers
Ant
Always a good sign, when you only need the smallest hint, and can otherwise do it all yourself.

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