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.
More to read
6.3 Taylor and Maclaurin Series
A Taylor series rewrites a smooth function as a power series built from its derivatives at one center. The coefficient of (x minus a) to the n comes from the nth derivative at a divided by n factorial. A Maclaurin series is the same idea centered at zero, which keeps the arithmetic simple. The classic examples are worth memorizing: e to the x, sin x, and 1 over (1 minus x) each have a clean pattern you can reuse. Once you know one series, substitution and term by term differentiation give you new ones for free. The remainder term then tells you how far a partial sum can be from the true value, so you know when to stop adding terms.
6.1 Power Series and Functions
A power series looks like an infinitely long polynomial, and inside its interval of convergence it behaves like a function. Adding, scaling, and matching coefficients work term by term, which is why series are so handy for approximation. The key question is always where the series converges, since outside that interval the partial sums stop meaning anything. Endpoints need separate checks with the usual convergence tests. This page builds the groundwork: once you trust power series as functions, Taylor polynomials become a natural next step rather than a trick.
6.2 Properties of Power Series
Inside the interval of convergence you can differentiate and integrate a power series term by term, and the interval stays the same. That single fact unlocks most Taylor series problems: start from one known series and operate on it. Multiplying a series by x or substituting a new expression into it follows the same term by term logic. These moves are exact, not approximations, as long as you stay inside the convergence interval. Read this page as a toolkit, because the Taylor section will use every tool here.
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16 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.