Chapter 9 AP Calculus BC

9.1 Power Series Infinite Series: Partial Sums: If the sequence of partial sums has a limit S, as n  infinity, then we say the series Converges to S. Otherwise it Diverges. Geometric Series: Interval of convergence : -1 < r < 1 Is the IOC for Geom. Series.

Representing functions by series: Definition of a Power Series: An expression of the form: is a Power Series centered at x=0. is a Power Series centered at x = a. Given:Find a Power Series for:

Again, Given: Find a Power Series for:Answer: Now, write down the series for: and use it to write one for :Answer: Copy down Theorem 1 p. 478 and Theorem 2 p. 479

9.2 Taylor Series Given:Find the Taylor Polynomial….. Answer: Construct a Power Series for:nth term answers: Taylor Series generated by f at x=0:

9.2 cont’d. p. 489 Taylor Series centered at x = a. Copy it down!!!! p most important series: Copy them down, know them!!! Occasionally you need 6, 7…….

9.3 Taylor’s Theorem Adequate substitution: a Taylor series that is off from the actual by less than Truncation Error: NEXT TERM!!!! If f has derivatives of all orders in an open interval, I, containing a, then for each positive integer, n, and for each x in the interval: Where:Largest value of derivative..f part Theorem 4 – Remainder Estimation Theorem. (do examples)

9.4 Radius of Convergence A convergent series is a number and may be treated as such…. R = radius of convergence and the set of x-values for which the series converges is called the Interval of Convergence. Theorem 6 – nth term test for divergence

9.4 cont’d. Direct Comparison Test (DCT) (non-negative terms) Greatest Power Rules!!!!! Theorem 8….. Ratio Test – (Powers and Factorials) Telescoping series: p. 510……

9.5 Testing Convergence at Endpoints P-series test:

9.5 cont’d. Alternating series Test (Liebniz’s Theorem) Error is next term sign included………… Look at examples 4 -6 pp Absolute and Conditional Convergence……