General Binominal Theorem

用戶195248

2007-10-13 9:02 pm

1. (a) Expand (1+x)^(2/3) in ascendin powers of x as far as the term containing x^3 for -1<1
(b) Using the expansion in (a), find the values of 81^(1/3)correct to 4 significant figures

2. (a) Expand (1-2x)^(1/3) in ascending powers of x as far as the term in x^3
(b) By taking x =1/8, evaluate 6^(1/3) correct to 2 decimal places using the expansion in (a)

回答 (1)

2007-10-14 7:36 am

✔ 最佳答案

Newton's generalized binomial theorem states that
(see http://en.wikipedia.org/wiki/Binomial_theorem)
(x+y)^n = sum k from 0 to infinity (Ck . x^k . y^(n-k) )
where Ck=n(n-1)...(n-k+1)/k!

1.a
So applying the formula,
(1+x)^(2/3)
= 1 + (2/3)x + (2/3)(-1/3)/2! x^2 + (2/3)(-1/3)(-4/3)/3! x^3 + ...
= 1 + (2/3)x - (1/9)x^2 + (4/81)x^3 + ...

1.b
Put x=1/8
(1+1/8)^2/3
=(9/8)^2/3
=(81/64)^1/3
=81^(1/3)/64^(1/3)
=81^(1/3)/4
= 11215/10368
Therefore the approximate value of
81^(1/3)
~= 4*11215/10368
=4.327 (to three decimals)
Check
4.327^3=81.014 OK

Question #2 is very similar to #1, and I am sure you would be capable to try it out as an exercise. If you have problems, post what you have done and we will take it from there.

👍 0 👎 0