ChatterBank1 min ago

# Maths again Exponential expansion

11 Answers

the question is

find a and b in termns of n if the expansions of (1+x)n and exp ( ax/(1+bx)) are the same up to x2

(1+x)n is easyy 1 +nx + n(n-1)/2! x2 etc

but the exponent : 1 + ax/1+bx + a2x2/(1+bx)2/2!...

still doesnt give ascending powers of x

so I am not able to equate and compare coefficients

please help (a level 1968)

find a and b in termns of n if the expansions of (1+x)n and exp ( ax/(1+bx)) are the same up to x2

(1+x)n is easyy 1 +nx + n(n-1)/2! x2 etc

but the exponent : 1 + ax/1+bx + a2x2/(1+bx)2/2!...

still doesnt give ascending powers of x

so I am not able to equate and compare coefficients

please help (a level 1968)

# Answers

If you look again carefully you will see that it is correct. Here is a link to a pdf file which shows how it works very clearly:

https://docs.google.c...yMzctYjYxMTRmMDAwMzRh

ht

11:16 Fri 06th Jan 2012

Write ax/(1+bx) as ax(1+bx)^-1 and expand as

ax(1-bx+b^2x^2+terms in x^3 or above)

so exp(ax(1-bx+b^2x^2+terms in x^3 or above)) is:

1+ax(1-bx+b^2x^2+terms in x^3 or above)+a^2x^2(1-bx+b^2x^2+terms in x^3 or above)^2

=1+ax-abx^2+a^2x^2+terms in x^3 or above.

If the x terms and x^2 terms in this series must be the same as those of

(1+x)^n=1+nx+n(n-1)x^2+terms in x^3 or above, then

a=n and n^2-nb=n(n-1)/2, so b=(n+1)/2 as long as n is not equal to 0.

You can see that this works for n=1 giving a=1 and b=1

and for n=2 giving a=2 and b=3/2

https://docs.google.c...yMzctYjYxMTRmMDAwMzRh

ALSO - you know I anwered this q in 1968 and got it out but couldnt remember how I had done it

also in getting me to think about it -for which I thank you - I saw that the alternative is to expand exp (ax(1-bx) to exp (ax - abx2) as exp(ax)exp(-abx2) as

(1+ax +a2x2/2.......)(1-abx2/2......) which lead to the same exp when multiplied out viz....

1 + ax + (a2/2-ab)x2

I thank you very very much and can asterisk your answer as the BEST as the toggle seems to be dead - manuy thanks again

## Related Questions

Sorry, we can't find any related questions. Try using the search bar at the top of the page to search for some keywords, or choose a topic and submit your own question.