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>Background

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 is the sum of the first n terms of the series. Do all series for which for all n converge? No! ex: doesn’t converge. It approaches zero too slowly. (Proof in hw) Ch. 1- Infinite Series>Convergence and Divergence

A.Geometric series Each term is multiplied by a fixed number to get the next term. ex: 1) ) 2 – – 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: ex: … Ch. 1- Infinite Series>Convergence and Divergence>Geometric 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>Alternating Series

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 ∑b n converges, then ∑a n converges absolutely. Thm: If a series converges absolutely, then it converges. (eg, if ∑b n converges, then ∑a n converges in above example.) Ch. 1- Infinite Series>Convergence and Divergence>More General Results>Preliminary 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>Comparison Test

3) Ratio Test: For this test, we compare a n+1 to a n : 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: ex: Harmonic Series Ch. 1- Infinite Series>Convergence and Divergence>More General Results>Ratio Test

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

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: converges by alternating series test. if x = 1: (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>Convergence

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) = e x 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 MacLaurin 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 MacLaurin Series expansion.) ex: sin(x 2 ) Ch. 1- Infinite Series>Power Series>Expanding Functions