SOLUTION: Expand (1/4+2x) as a series of ascending powers of x up to the term x^3 and state the range of values of x for the expansion to be valid. Hence, use the expansion to approximate 1/

Algebra ->  Test -> SOLUTION: Expand (1/4+2x) as a series of ascending powers of x up to the term x^3 and state the range of values of x for the expansion to be valid. Hence, use the expansion to approximate 1/      Log On


   

Question 1184848: Expand (1/4+2x) as a series of ascending powers of x up to the term x^3 and state the range of values of x for the expansion to be valid. Hence, use the expansion to approximate 1/(4.05)^2 correct to 4 significant figures by taking a suitable value of x
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
I will interpret you function as f%28x%29+=+1%2F%284%2B2x%29, since the function 1%2F4+%2Bx won't have a Taylor (Maclaurin) series that will even reach x%5E3 term.
It is straightforward exercise that 1%2F%284%2B2x%29+=+sum%28%28-1%29%5En%2A%28x%5En%2F2%5E%28n%2B2%29%29%2C+n=0%2C+infinity%29.
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Up to the x%5E3 term, you get f%28x%29+=+1%2F%284%2B2x%29%5E2~.
Letting x = 0.025, f%28x%29+=+1%2F%284.05%29%5E2~+1%2F16-0.025%2F16%2B%283%2F64%290.025%5E2-%281%2F32%290.025%5E3+=+%220.0609663...%22, or 0.0610 to 4 significant figures.
The Taylor expansion is valid for all abs%28x%29+%3C+2, by the ratio test.