Taylor and Maclaurin Series
Build the power series of a function from its derivatives at a point, use the remainder to say how good a truncation is, and reuse the standard expansions instead of starting over.
What a learner can do afterwards
- Derive the Maclaurin series of e^x, sin x or 1/(1 - x)
- Obtain a new series by substituting into or differentiating a known one
- Bound the error of a Taylor polynomial with the remainder term
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A Taylor series rebuilds a smooth function from its derivatives at one center. The coefficient of (x minus a) to the n is the nth derivative at a divided by n factorial. A Maclaurin series is the same idea centered at 0, which keeps the arithmetic simple. For e to the x, every derivative at 0 equals 1, so you get 1 + x + x squared over 2 and on.
For sin x the derivatives at 0 cycle through 0, 1, 0, minus 1, so the even terms vanish and degree 3 gives x minus x cubed over 6. For 1 over (1 minus x) every derivative at 0 equals n factorial, so dividing by n factorial leaves coefficient 1 on every power: 1 + x + x squared and on. Stopping after a chosen term gives a Taylor polynomial, like T2(x) = 1 + x + x squared over 2 for e to the x.
Once you know one series you get new ones for free by substitution or term by term moves, valid inside the interval. Replacing x with minus x flips every odd power, turning e to the x into e to the minus x. Multiplying each term by x gives the series for x times e to the x. Differentiating term by term works the same way.
The Lagrange remainder bounds your error with the largest value of the next derivative on the interval, times the power term over (n + 1) factorial. When that bound is tiny, your partial sum is trustworthy. Use it to certify approximations like cos(0.1) before you stop adding terms.
Derivatives at the center give the coefficients, known series give new ones, and the remainder certifies the stop.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.