Ch 1 Infinite SeriesBackground Chapter 1 Infinite Series

Ch. 1 - Infinite Series>Background Chapter 1: Infinite Series I. Background • An infinite series is an expression of the form: where there is some rule for how the a’s are related to each other. ex:

Ch. 1 - Infinite Series>Background • Why do physicists care about infinite series? 1) Loads of physics problems involve infinite series. ex: Dropped ball- how far does it travel? ex: Swinging pendulum- how long until it stops swinging? (or will it ever stop? ) *picture? *

Ch. 1 - Infinite Series>Convergence and Divergence 2) Complicated math expressions can be approximated by series and then solved more easily. ex: II. Convergence and Divergence • How do we know if a series has a finite sum? (eg. will the pendulum ever stop? ) defn: Mathematics terminology- The series converges if it has a finite sum; otherwise, the series diverges. defn: We define the sum of a series (if it has one) to be: where • Do all series for which ex: is the sum of the first n terms of the series. for all n converge? No! doesn’t converge. It approaches zero too slowly. (Proof in hw)

Ch. 1 - Infinite Series>Convergence and Divergence>Geometric Series A. Geometric series • Each term is multiplied by a fixed number to get the next term. ex: 1) 1 + 3 + 9 + 27 +. . . 2) 2 – 5 + 18 – 54 +. . . • We can show that only for a geometric series, the sum of the first n terms is Proof: (geometric series only)

Ch. 1 - Infinite Series>Convergence and Divergence>Geometric Series The sum of the geometric series is then: (for |r|<1, geometric series only) ex: 0. 583333…

Ch. 1 - Infinite Series>Convergence and Divergence>Alternating Series B. Alternating Series: Series whose terms are alternately positive and negative. ex: • Test for converging for alternating series: An alternating series converges if the absolute value of the terms decreases steadily to zero. and ex:

Ch. 1 - Infinite Series>Convergence and Divergence>More General Results>Preliminary Test C. More general results: There are loads of other types of series besides geometric and alternating. So, how do we find whether a general series converges? This is a hard problem. Here are some simple tests (tons more exist). We’ll look at 3 tests: 1) Preliminary Test: If the terms of an infinite series do not tend to zero , then the series diverges. Note: this test does not tell you whether the series converges. It only weeds out wickedly divergent series. ex:

Ch. 1 - Infinite Series>Convergence and Divergence>More General Results>Preliminary Test The next tests are for convergence of series of positive terms, or for absolute convergence of a series with either all positive or some negative terms. defn: Say we have a series (series #1) with some negative terms. Then say we make a new series (series #2) by taking the absolute value of each term in the original series. If series #2 converges, then we say series #1 converges absolutely. ex: If ∑bn converges, then ∑an converges absolutely. Thm: If a series converges absolutely, then it converges. (eg, if ∑bn converges, then ∑an converges in above example. )

Ch. 1 - Infinite Series>Convergence and Divergence>More General Results>Comparison Test 2) Comparison Test: a) Compare your series a 1+a 2+a 3+… to a series known to converge m 1+m 2+m 3+…. If for all n from some point on, then the series a 1+a 2+a 3+… is absolutely convergent. b) Compare your series a 1+a 2+a 3+… to a series known to diverge d 1+d 2+d 3+…. If for all n from some point on, then the series a 1+a 2+a 3+… is divergent. ex: does this converge?

Ch. 1 - Infinite Series>Convergence and Divergence>More General Results>Ratio Test 3) Ratio Test: For this test, we compare an+1 to an: in the limit of large n: Ratio test: If p < 1, the series converges. If p = 1, use a different test. If p > 1, the series diverges. ex: Harmonic Series

Ch. 1 - Infinite Series>Power Series III. Power Series defn: A power series is of the form: where the coefficients an are constants. Note: Commonly, we see power series with a=0: ex:

Ch. 1 - Infinite Series>Power Series>Convergence A. Convergence of a power series depends on the values of x. m ex:

Ch. 1 - Infinite Series>Power Series>Convergence We must consider the endpoints ± 1 separately: (because these points fail the ratio test) ? ? ? keep the following ? ? if x = -1: if x = 1: converges by alternating series test. (harmonic series), so it diverges at x=1. Thus, our power series converges for -1≤ x <1 and diverges otherwise.

Ch. 1 - Infinite Series>Power Series>Expanding Functions B. Expanding functions as power series: From the previous section, we know that the sum of a power series depends on x: So, S(x) is a function of x! Useful trick: Try to expand a given function f(x) as a power series (Taylor series. ) (We often do this when the original function is too complex to use easily. ) ex: f(x) = ex

Ch. 1 - Infinite Series>Power Series>Expanding Functions More generally: How do we find the Taylor Series expansion of a general function f(x): (This approximates f(x) near the point x=a. ) Here’s how: Evaluating each of these at x=a: So, our Taylor series expansion of f(x) about the point x=a is:

Ch. 1 - Infinite Series>Power Series>Expanding Functions defn: a Mac. Laurin Series is a Taylor Series with a=0. ex: f(x) = sin(x) ex: Electric field of a dipole

Ch. 1 - Infinite Series>Power Series>Expanding Functions And, you can do all sorts of math with these series to get other series… (see section 13 for examples) ex: (x 2+3) sin(x) (find the Mac. Laurin Series expansion. ) ex: sin(x 2)