Representation of functions by Power Series We will use the familiar converging geometric series form to obtain the power series representations of some elementary functions. If ırı < 1, then the power series: The sum of such a converging geometric series is: Conversely, if an elementary function is of the form: Its power series representation is: By identifying a and r, such functions can be represented by an appropriate power series.

Find a geometric power series for the function: Make this 1 Divide numerator and denominator by 4 a = 3/4r = x/4 Use a and r to write the power series.

Find a geometric power series for the function: Let us obtain the interval of convergence for this power series. -4 < x < 4 f(x) Watch the graph of f(x) and the graph of the first four terms of the power series. The convergence of the two on (-4, 4) is obvious.

Find a geometric power series centered at c = -2 for the function: Make this 1 Divide numerator and denominator by 6 a = ½r – c = (x + 2)/6 The power series centered at c is: x = -2 f(x)

Find a geometric power series centered at c = 2 for the function: Since c = 2, this x has to become x – 2 Add 4 to compensate for subtracting 4 Divide by 3 to make this 1 Make this – to bring it to standard form

Find a geometric power series centered at c = 2 for the function: This series converges for: x = 2

Find a geometric power series centered at c = 0 for the function: Make this – to bring it to standard form We first obtain the power series for 4/(4 + x) and then replace x by x 2 Replace x by x 2 f(x)

Find a geometric power series centered at c = 0 for the function: We obtain power series for 1/(1 + x) and 1/(1 – x) and combine them. f(x) STOP HERE!

Find a geometric power series centered at c = 0 for the function: We obtain power series for 1/(1 + x) and obtain the second derivative. Replace n by n + 2 to make index of summation n = 0

Find a geometric power series centered at c = 0 for the function: We obtain power series for 1/(1 + x) and integrate it Then integrate the power series for 1/(1 – x ) and combine both.

Find a geometric power series centered at c = 0 for the function: When x = 0, we have 0 = 0 + cc = 0

Find a geometric power series centered at c = 0 for f(x) = ln(x 2 + 1) We obtain the power series for this and integrate. When x = 0, we have0 = 0 + c c = 0 -1 < x < 1 f(x)

Find a power series for the function f(x) = arctan 2x centered at c = 0 We obtain the power series for this and integrate. Replace x by 4x 2

Find a power series for the function f(x) = arctan 2x centered at c = 0 When x = 0, arctan 2x = 00 = 0 + cc = 0 f(x) = arctan 2x -0.5