A quotient can be written as a product with a negative exponent: sqrt(1+ax)1-bx=(1+ax) 12(1-bx)-1. Expand each factor to the required degree, multiply systematically and collect like powers. If the multiplier contains only non-negative powers, terms of higher degree cannot contribute to a lower-degree coefficient. If P=1+ax+bx2+. s and Q=1+cx+dx2+. s, then PQ=1+(a+c)x+(b+ac+d)x2+. s. After cancellation, distinguish the first three non-zero terms from all terms up to x3. They need not describe the same truncation. For a general coefficient in (1-kx)-2, use (1-kx)-2= _r=0infinity(r+1)kr xr, valid for |kx|<1.

A quotient can be written as a product with a negative exponent: \(\frac{\sqrt{1+ax}}{1-bx}=(1+ax)^{\tfrac{1}{2}}(1-bx)^{-1}\).

Expand each factor to the required degree, multiply systematically and collect like powers. If the multiplier contains only non-negative powers, terms of higher degree cannot contribute to a lower-degree coefficient.

If \(P=1+ax+bx^2+\cdots\) and \(Q=1+cx+dx^2+\cdots\), then \(PQ=1+(a+c)x+(b+ac+d)x^2+\cdots\).

After cancellation, distinguish the first three non-zero terms from all terms up to \(x^3\). They need not describe the same truncation.

For a general coefficient in \((1-kx)^{-2}\), use \((1-kx)^{-2}=\sum\limits_{r=0}^{\infty}(r+1)k^r x^r\), valid for \(|kx|<1\).

Binomial Theorem

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Combining binomial series

Combining binomial series