Taylor and Maclaurin Series: 20 flashcards
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1. What is a Taylor series?
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An infinite sum of powers of $(x-a)$ that represents a smooth function using its derivatives at the centre $a$: $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n.$$
2. What is the centre (expansion point) of a Taylor series?
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The fixed point $a$ about which the series is built; all derivatives are evaluated at $a$.
3. What is a Maclaurin series?
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The Taylor series with centre $a=0$: $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n.$$
4. Why does $n!$ appear in the denominator of every Taylor term?
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Differentiating $(x-a)^n$ repeatedly produces $n!$ as a leftover constant, so dividing by $n!$ cancels it and makes the $n$-th derivative match $f^{(n)}(a)$.
5. State the general Taylor series formula.
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$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$$
6. State the Maclaurin series formula.
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$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$$
7. What is the Maclaurin expansion of $e^x$?
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$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$
8. What is the Maclaurin expansion of $\sin x$?
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$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$
9. What is the Maclaurin expansion of $\cos x$?
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$$\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$$
10. What is the remainder term $R_n(x)$?
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The error left after truncating the series after $n$ terms; in Lagrange form, $$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$$ for some $c$ between $a$ and $x$.
11. What is the interval of convergence?
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The set of $x$ values for which the Taylor series actually converges to the function; outside it, the series is not valid.
12. For which $x$ does the Maclaurin series of $\frac{1}{1-x}$ converge?
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Only for $|x|<1$.
13. Which functions have Maclaurin series that converge for all real $x$?
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$e^x$, $\sin x$, and $\cos x$ all converge for every real $x$.
14. List the steps to build a Taylor series about $a$.
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1. Compute $f(a)$. 2. Compute $f'(x)$, then $f'(a)$. 3. Compute $f''(x)$, then $f''(a)$, and continue. 4. Substitute into $\sum \frac{f^{(n)}(a)}{n!}(x-a)^n$.
15. Where are Taylor polynomials used in real work?
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Calculators and computers evaluate $\sin$, $\cos$, and $e^x$ using truncated Taylor polynomials; physicists linearise equations using the first one or two terms.
16. What is the small-angle approximation and where does it come from?
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$\sin\theta \approx \theta$, which comes from keeping only the first term of the Maclaurin series of $\sin\theta$.
17. Common mistake: what happens if you forget the factorial in the denominator?
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The series no longer matches the derivatives of the original function, and the approximation collapses.
18. Common mistake: what goes wrong when the centre is not zero?
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Students plug $x$ into $f^{(n)}(a)$ instead of evaluating every derivative at $a$. If $a=2$, you need $f'(2)$, $f''(2)$, and so on — not $f'(x)$ left as a formula.
19. Why is a Taylor series useful for computation?
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It replaces complicated functions like $e^x$, $\sin x$, or $\ln(1+x)$ with polynomials that are easy to add, multiply, differentiate, and integrate, and behave almost identically near the chosen point.
20. How does each term of a Taylor series relate to the original function's derivatives?
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Each term is engineered so that the $n$-th derivative of the series at $x=a$ equals $f^{(n)}(a)$ exactly, keeping one more derivative in agreement with $f$.
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