Ch 9.1 Power Series Calculus Graphical, Numerical, Algebraic by Finney, Demana, Waits, Kennedy

What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate Calculators use polynomials to calculate trig functions, logarithmic functions etc. Downfall of polynomial equivalent functions is that they have an infinite number of terms.

For Example Power Series: an infinite sum of variables to a power. y = sin (x) can be represented as a power series: Every time you add a term to the series it fits sin x even better. Let’s check this out using the calculator and then the geometer sketchpad.

Power Series for cos x If sin x can be represented by the power series: Then cos x can be represented by the power series derived from taking the derivative of sin x: Let’s check it out on the calculator…

Power Series for cos x

Geometric Series Partial Sum of a Geometric Series: Sn = a + ar + ar2 + ar3 + … + arn-1 -[r Sn = ar + ar2 + ar3 + … + arn ] Sn – r Sn = a + arn Sn (1 – r) = a (1 - rn)

Sum of Converging Series

Power Series Using Calculator

Example of a Power Series

Convergent Series Only two kinds of series converge: 1) Geometric whose | r | < 1 2) Telescoping series Example of a telescoping series: the middle terms cancel out

Graph both of these functions on your calculator!

Now graph both y = ln (1+x) And the power series below and check the fit!

Finding Power Series for Functions Given that 1/(1-x) is represented by the power series: 1 + x + x2 + … + xn + … On the interval (-1,1), Find a power series that represent 1/(1+x) on (-1,1). Find a power series that represents x/(1+x) on (-1,1). Find a power series that represents 1(1-2x) on (- ½ , ½ ) Find a power series that represents and give its interval of convergence

Finding Power Series for Functions Given that 1/(1 - x) is represented by the power series: 1 + x + x2 + … + xn + … on the interval (-1,1), Find a power series that represent 1/(1+x) on (-1,1) 1 - x + x2 – x3 … + (-x)n + … 2. Find a power series that represents x/(1+x) on (-1,1). x – x2 + x3 – x4 + x5 – x6…. + (-1)n xn+1 + … 3. Find a power series that represents 1/(1 - 2x) on (- ½ , ½ ) 1 + 2x + 4x2 + 8x3 + … + (2x)n + … Find a power series that represents 1 – (x-1) + (x-1)2 – (x-1)3 + … + (-1)n (x-1)n + … 5. Find a power series that represents

Finding a series for tan-1 x 1. Find a power series that represents on (-1,1) Use integration to find a power series that represents tan-1 x. Graph the first four partial sums. Do the graphs suggest convergence on the open interval (-1, 1)? 4. Do you think that the series for tan-1 x converges at x = 1?

Finding a series for tan-1 x 1. Find a power series that represents on (-1,1) Use integration to find a power series that represents tan-1 x. Graph the first four partial sums. Do the graphs suggest convergence on the open interval (-1, 1)? yes 4. Do you think that the series for tan-1 x converges at x = 1? Yes to

Guess the function Define a function f by a power series as follows: Find f ‘(x). What function is this?

Guess the function Define a function f by a power series as follows: Find f ‘(x). What function is this? ex