Chapter 1: Infinite Series I. Background An infinite series is an expression of the form:

1. 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:

2. 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?*

3. 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 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)

4. Ch. 1- Infinite Series>Convergence and Divergence>Geometric Series • 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)

5. 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: 0.583333…

6. 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:

7. 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: • 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:

8. 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.)

9. Ch. 1- Infinite Series>Convergence and Divergence>More General Results>Comparison Test 2) Comparison Test: a) Compare your series a1+a2+a3+… to a series known to converge m1+m2+m3+…. If for all n from some point on, then the series a1+a2+a3+… is absolutely convergent. b) Compare your series a1+a2+a3+… to a series known to diverge d1+d2+d3+…. If for all n from some point on, then the series a1+a2+a3+… is divergent. ex: does this converge?

10. 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: ex: Harmonic Series

11. 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:

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

13. 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.

14. 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

15. 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:

16. 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

17. 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: (x2+3) sin(x) (find the MacLaurin Series expansion.) ex: sin(x2)