How Do You Find the Maclaurin Series for e^x, sin x, cos x, and ln(1+x)?

Use the Maclaurin series formula

$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n.$$

Find successive derivatives, evaluate them at $x=0$, identify the coefficient pattern, and substitute into the formula. This is IB Mathematics: Analysis and Approaches HL content, specifically AHL 5.19, Maclaurin series.

How the patterns arise

For $\ln(1+x)$, begin with the geometric series

$$\frac{1}{1+x}=1-x+x^2-x^3+\cdots.$$

Integrating term by term from $0$ to $x$ gives

$$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots.$$

Its interval of convergence is $-1<x\leq 1$. By contrast, the series for $e^x$, $\sin x$, and $\cos x$ converge for every real .

For example, four terms give

$$e^{0.2}\approx 1+0.2+\frac{0.2^2}{2!}+\frac{0.2^3}{3!}=1.22133.$$

The exact value is approximately $1.22140$, so this truncated series is already accurate near $x=0$. More terms generally improve the approximation within the interval of convergence.

A systematic derivative table is useful whenever the required series is not supplied. Write the function and several successive derivatives in separate rows, then evaluate every row at zero. This makes zero coefficients, alternating signs, and repeating cycles visible before any substitution. It also prevents an indexing error when converting the pattern into sigma notation. After obtaining a series, check its constant term against the original function's value at zero and differentiate the first few terms to confirm the pattern.

Exam technique

In an AA HL response, show the derivative cycle or state the known series before writing the required expansion. For a show question, display enough coefficient calculations to establish the result; simply copying the final series may not earn all method marks.

A common misconception is omitting factorials or mixing odd and even powers. Remember: $\sin x$ uses odd powers, $\cos x$ uses even powers, and both alternate signs. Also distinguish a finite Maclaurin polynomial from the infinite series by using an approximation sign for truncation. These methods may be tested analytically on Paper 1 or through substitution, products, differentiation, and integration on Papers 2 and 3.