How Do You Use Substitution to Find the Maclaurin Series of a Related Function?

Use a known Maclaurin series, replace its variable with the required expression, and simplify the resulting powers of $x$. Also transform any convergence condition and retain enough terms to reach the requested order.

The Substitution Method

This method belongs to AHL 5.19, Maclaurin series in IB Mathematics: Analysis and Approaches HL. It may be tested analytically on Paper 1 or Paper 2, and it avoids recalculating every coefficient from the general Maclaurin formula.

Suppose a known expansion is

$$f(u)=a_0+a_1u+a_2u^2+a_3u^3+\cdots.$$

To find the expansion of (f(g(x))), substitute $u=g(x)$:

$$f(g(x))=a_0+a_1g(x)+a_2(g(x))^2+a_3(g(x))^3+\cdots.$$

Then expand and collect like powers of $x$.

The key idea is that composition changes both the powers and the number of source terms required. If $g(x)$ begins with $x^k$, then $u^n$ begins with $x^{kn}$. Therefore, to obtain terms through , keep source terms only while . When contains several terms, expand each power carefully and collect like powers; higher source powers may sometimes still contribute to the requested order.

For example, begin with

$$e^u=1+u+\frac{u^2}{2!}+\frac{u^3}{3!}+\cdots.$$

Substituting $u=x^2$ gives

$$e^{x^2}=1+x^2+\frac{x^4}{2!}+\frac{x^6}{3!}+\cdots =1+x^2+\frac{x^4}{2}+\frac{x^6}{6}+O(x^8).$$

Therefore, this is the expansion up to and including the term in $x^6$. Notice that substituting $x^2$ doubles every exponent, so only the first four source terms are needed.

For a restricted series,

$$\ln(1+u)=u-\frac{u^2}{2}+\frac{u^3}{3}-\cdots,\qquad \lvert u\rvert<1.$$

Putting $u=2x$ produces

$$\ln(1+2x)=2x-2x^2+\frac{8x^3}{3}-\cdots,$$

with $\lvert 2x\rvert<1$, or $\lvert x\rvert<\frac{1}{2}$.

A common misconception is to leave the original convergence condition unchanged. The condition must undergo the same substitution as the series.

Exam Technique

For the command term find, show the known series, substitution, simplification, and final order term. Do not give only the final expansion: explicit working is needed for method marks.