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[h] IB Mathematics AA HL Flashcards -Maclaurin series
[q] Maclaurin series
Power Series:
This is an infinite series of multiples of powers of x, that can surprisingly be used to, quite effectively, approximate all kinds of functions that seemingly are unrelated to polynomials. Below is an example (for sinx) that shows how the power series increases in accuracy as you add terms:
\[ \sin(x) = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots \]
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Derivation:
So how did we come up with that previously stated power series for sinx? Specifically, \( f(x) = \sin(x) \), centered at zero?
1. We start with the premise of getting a polynomial, that has the same value and slope as \( f(x) = \sin(x) \) at \( x = 0 \). i.e. \( f(0) = P(0) \) & \( f'(0) = P'(0) \). For this first step, we can just do this with a polynomial in the form \( P(x) = a_0 + a_1x \), then:
\[
f(0) = \sin(0) = 0, f'(0) = \cos(0) = 1, P(0) = a_0, P'(0) = a_1 \Rightarrow a_0 = 0, a_1 = 1
\]
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This gives us \( P(x) = 0 + 1x \Rightarrow P(x) = x \). We clearly need more terms. Right now, we only have a straight line, but it does have the right gradient at \( x = 0 \). If we also had the correct 2nd derivative, and the same 3rd, 4th, 5th, … nth derivatives as sinx, it would become a curve that keeps improving its approximation:
For 2nd & 3rd derivatives:
\( f”(0) = – \sin(0) = 0, f^{(3)}(0) = – \cos(0) = -1 \)
If \( P(x) = 0 + x + a_2x^2 \), \( P'(x) = 2a_2 \), \( a_2 = 0 \)
If \( P(x) = 0 + x + 0x^2 + a_3x^3 \), \( P^{(3)}(x) = 6a_3, 6a_3 = -1, a_3 = -\frac{1}{6} \)
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Carrying on this process infinitely, we get the series:
\[ \sin(x) = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots \]
Maclaurin Series:
To do this process for any function, we can generalize as follows:
\[
f(x) = f(0) + x f'(0) + \frac{x^2}{2!}f”(0) + \frac{x^3}{3!}f^{(3)}(0) + \cdots
\]
This is called the Maclaurin series, and is technically just a special case of the Taylor series — which is the same process, but centered at any point *c*, not just zero.
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Method: For finding a few terms for the Maclaurin series is as follows:
1. Find \( f'(x), f”(x), f^{(3)}(x), \ldots \)
2. Evaluate \( f'(0), f”(0), f^{(3)}(0), \ldots \)
3. Plug into the Maclaurin series formula
Given Series:
The following Maclaurin series are just given in the formula booklet, so you won’t have to calculate them manually:
\[
e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
\]
\[
\ln(1 + x) = x – \frac{x^2}{2} + \frac{x^3}{3} – \cdots
\]
\[
\sin(x) = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \cdots
\]
\[
\cos(x) = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \cdots
\]
\[
\tan^{-1}(x) = x – \frac{x^3}{3} + \frac{x^5}{5} – \cdots
\]
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