Representation of functions by Power Series We will

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 = 3/4 r = 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. f(x) -4 < x < 4 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: The power series centered at c is: Divide numerator and denominator by 6 a=½ Make this 1 r – c = (x + 2)/6 f(x) x = -2

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

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: We first obtain the power series for 4/(4 + x) and then replace x by x 2 Replace x by x 2 Make this – to bring it to standard form 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+c c=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. f(x) When x = 0, we have 0=0+c c=0 -1 < x < 1

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

Find a power series for the function f(x) = arctan 2 x centered at c = 0 f(x) = arctan 2 x -0. 5 When x = 0, arctan 2 x = 0 0=0+c -0. 5 c=0