use binomial series to find a maclaurin series?

Question

for (4+x^2)^(1/2) or sqrt(4+x^2)
write at least 6 terms...

Answer 1

The Maclaurin series is the Taylor series, expanded about x = 0. The terms of the series are f(0) + f'(0)*x / 1! + f''(0)*x^2 / 2! + f'''(0)*x^3 / 3! + ...

So, look at the first few terms. f(x) = sqrt(4 + x^2), f'(x) = x / sqrt(4 + x^2), and f''(x) = (sqrt(4 + x^2) - (x^2 / sqrt(4 + x^2))) / sqrt(4 + x^2). The derivatives get messier as we go on, but you do need to find three more derivatives to get six terms. Dealing with just the first three terms, f(0) = 2, f'(0) = 0, and f''(0) = 1. So the first three terms are 2 + 0x + x^2, which simplifies to 2 + x^2, which is technically only two terms. You should find out from your teacher whether or not terms with a coefficient of zero count; if they don't, you'll need to calculate even more derivatives to get non-zero terms.