Recover polynomial coefficients from derivatives at zero. Verify eigenvectors and use a diagonal form to compute a matrix power. Add a particular integral to the complementary function. Use a finite geometric formula and then take real or imaginary parts.
Notes and key methods
A Maclaurin expansion starts \(\displaystyle f(x)= f(0)+ f'(0)x+ \frac{f''(0)x^{2}}{2!}+ \frac{f^{(3)}(0)x^{3}}{3!}+ \cdots\). Each coefficient includes its factorial denominator.
For a product, multiply truncated series while collecting every contribution to a given power. A truncated expression is an approximation away from zero; state the retained order rather than treating it as an exact identity.
An eigenvector \(\displaystyle v\ne 0\) satisfies Av=λv. If a square matrix has a basis of eigenvectors, form Q from those vectors as columns and D from the corresponding eigenvalues in the same order: A=QDQ⁻¹.
Then \(\displaystyle A^{n}=QD^{n}Q^{−1}\), and the trace and determinant are respectively the sum and product of eigenvalues. Distinct real eigenvalues in these examples guarantee a real eigenvector basis.
For a linear constant-coefficient equation, the general solution is the complementary function plus one particular integral. An exponential forcing Keᶜˣ normally suggests \(\displaystyle Ae^{cx}\).
If c is an auxiliary root, multiply the trial by x for a simple root, or by \(\displaystyle x^{2}\) for a repeated root. Resonance concerns overlap with the complementary solution, not merely a large response. Determine remaining constants after adding the particular integral.
For \(\displaystyle z\ne 1\), \(\displaystyle \sum_{k= 0}^{n- 1} z^{k}= \frac{z^{n}- 1}{z- 1}\). Write z in polar form, then take real or imaginary parts to obtain corresponding weighted cosine or sine sums.
The initial power matters: a sum starting at \(\displaystyle k=1\) is z times the sum starting at \(\displaystyle k=0\). For \(\displaystyle z=1\) use n directly, rather than dividing by zero in the geometric formula.
Check the conditions
The required derivatives exist near zero.
State the order to which the series is truncated.
An eigenvector is nonzero.
Match the order of Q columns to the order of diagonal entries in D.
Check the forcing trial against the auxiliary roots.
Substitute the trial to determine its coefficient.
The quotient formula requires \(\displaystyle z\ne 1\).
State whether the sum begins with 1 or with z.
Three worked examples
Example 1 · Maclaurin expansions
Let \(f(x)=e^{x} \cos{\left(2 x \right)}\). Write \(f(x)=1+Ax+Bx^2+Cx^3+O(x^4)\).
Find A.
Find B.
Find C.
Use the cubic polynomial to approximate \(\displaystyle f(\frac{1}{25})\).