The function f(x) = `sqrt(x + 2) is to be expanded about a = -1.

• The first three derivatives are easily calculated as shown below, and their values at a = -1 are indicated.

Video Clip #1

The Taylor polynomial of degree three for f(x) = `sqrt(x) about x = 1 is indicated in the second line below.

• The Taylor polynomial for f(x+2) = `sqrt(x+2) is easily found by substituting x+2 for x in the original Taylor polynomial.
• The resulting Taylor polynomial is expressed in powers of (x + 1), and so constitutes an expansion of `sqrt(x+2) about x = -1.
• Since a graphical picture shows that `sqrt(x+2) is just `sqrt(x) shifted 2 units to the left, it makes sense that the value of a about which the sequence is expanded is also shifted 2 units to the left, from a = 1 to a = -1.

Video Clip #2

In the figure below we directly obtain the degree-4 expansion of f(x) = cos(2x) about a = `pi / 8.

• The first four derivatives are as indicated.
• The values of these derivatives at a = `pi / 8 are calculated and substituted into the form of the Taylor polynomial.
• The Taylor polynomial will be expressed in powers of (x - `pi / 8), as indicated.

If we wished to use the expansion of cos(2x) to find expansion for cos(x), we could find f(x/2).

• The Taylor polynomial would then give us an expansion about a = `pi / 4, as we see from the algebra in the figure below.
• A power of (x / 2 - `pi / 8) is a power of 1/2 (x - `pi/4); by incorporating the power of the factor 1/2 into the coefficient, we obtain a power of (x - `pi / 4).
• It is easy to verify that the final result is identical to the expansion we would get for cos(x) using the Taylor formula.
• Thus when f(x) represents a Taylor expansion about some value a, f(x/2) will represent a Taylor expansion about 2a; more generally f(kx) will represent a Taylor expansion about a / k.

Video Clip #3

The series in the figure below has coefficients 1, 1/4, 1/9, 1/16, ..., 1/n^2.

• It follows that | C(n+1) | / | Cn | = (n / (n+1) ) ^2 has a limit of 1 as n -> infinity.
• It follows that the series has radius a convergence R = 1.