MATH 163 — R E V I E W E X A M 3 — FALL 2016

TOPICS COVERED 1. Sequences {a }∞ n n=n0 ◦ Find limits of sequences ◦ Graph sequences ∞ P 2. Series an n=n0 ◦ Know the definition of the value of a series in terms of the limit of partial sums. Distinguish between the sequence of summands {a }∞ , and the sequence of partial sums {s }∞ , n n=n0 n n=n0 where s = Pn a . n k=n0 k ◦ Define divergence, convergence, and absolute convergence of a series. Know examples of series that diverge, converge absolutely, and converge conditionally. ◦ Be able to evaluate converging - geometric series (this is the Taylor series for f(x) = 1/(1 − x)) - telescoping series - series that happen to be other known Taylor series for functions such as ex, cos(x), or sin(x), at particular values of x. P∞ n P∞ n P∞ n Examples: n=0 1/2 , n=0 (−10) /n!, n=0 n(x − 1) , ◦ Be able to determine whether a series diverges, converges absolutely, or converges condition- ally, using the Divergence Test, Limit Comparison Test, Direct Comparison Test, Ratio Test, Integral Test, Alternating Series Test. ◦ Be able to estimate series by partial sums using the alternating series estimation test. Do you know of any other methods to estimate series?

∞ P n 3. Power Series cn(x − a) , power series representation of functions n=0 ◦ Find radius and interval of convergence of a power series. P∞ n ◦ Uniqueness. The power series representation n=0 cn(x − a) of a function about the base point x = a is unique: if f has a power series representation about x = a that converges to f in (a − R, a + R), for some R > 0, then the unique coefficients are given by

f (n)(a) c = n n! 4. Taylor series ◦ Know how to find Taylor series for any function either using the definition

∞ X f (n)(c) f 00(c) f 000(3) (x − a)nf(c) + f 0(c)(x − a) + (x − a)2 + (x − a)3 + ... n! 2 3! n=0

or using known Taylor series (e.g. for 1/(1 − x), ex, sin(x), cos(x) and substitution addition, multiplication differentiation integration ◦ When can you add, multiply, differentiate, integrate a series term-by-term? ◦ Memorize the Taylor series about x = 0 (the MacLaurin series) for

1 f(x) = , f(x) = ex , f(x) = sin(x) , f(x) = cos(x) , 1 − x

and their intervals of convergence.

◦ Find Taylor polynomials pn(x) of degree n for given function f, about a base point x = a. ◦ Be able to approximate a function by a Taylor polynomial f(x) ≈ pn(x) and estimate the error Rn(x) made using - Taylor’s Inequality. - the Alternating Series Estimation test. (n+1) f (c) n+1 Note: f(x) = pn(x) + Rn(x) where Rn(x) = (n+1)! (x − a) , where c is some (unknown) number between x and a. This holds, provided f is n + 1 times continuously differentiable between x and a. This formula is a generalization of the Mean Value Theorem, and we need it in order to determine how good the approximation of a function by a polynomial is.

◦ Note: the Taylor polynomial pn(x) for f(x) about x = a is the unique polynomial of degree n that satisfies that all its derivatives of order ≤ n agree with the derivatives of f at x = a,

(k) (k) pn (a) = fn (a) , k ≤ n

For example, p1(x) is our already well-known linear approximation L(x) of f(x), which satisfies that L(a) = f(a) and L0(a) = f 0(a).

PRACTICE PROBLEMS The topics listed above form a summary of the material you need to know. Read it and make sure you understand the main ideas. A set of sample problems is given below.

1. (a) What does it mean to state that limk→∞ ak = L? P∞ (b) What does it mean to state that k=1 ak = L? P∞ Pn (c) Assume k=1 ak = L. What is limk→∞ ak? What is limn→∞ sn, where sn = k=1 ak? k (d) For ak = 1/2 , (i) sketch a plot of the sequence {ak} Pn (ii) sketch a plot of the sequence {sn}, where sn = k=1 ak? P∞ (iii) Find k=1 ak.

3. What is a telescoping series? Give two examples of telescoping series and determine their value, or determine that they diverge.

4. Consider the geometric series P∞ rk. Give two examples of geometric series, one starting k=k1 at k1 = 0, another starting at k1 6= 0, and find their values.

5. What is an absolutely converging series? What is a conditionally converging series? Give an example of each.

6. Fill in the blanks using either may or must. Explain your answers. (a) A series with summands tending to 0 converge. (b) A series that converges have summands that tend to zero. (c) If a series diverges, then the summands not tend to 0. (d) If a series diverges, then the Divergence Test succeed in proving the divergence. P∞ P∞ (e) If n=10 an diverges, then n=1000 an diverge. P∞ P∞ (f) If n=10 an converges, then n=10 |an| converge. P∞ P∞ (g) If n=10 |an| converges, then n=10 an converge. 7. (a) REVIEW Chapter 11 (pp 802), Concept Check: 9 ∞ X xn (b) Let f(x) = . Find series representations for f 0, f 00. Find the intervals of conver- n2 n=1 gence of f, f 0, and f 00. (Make sure you use your answer to (a)!)

8. REVIEW Chapter 11 (pp 802-804), True-False: 6,9,10,11,17,18,19. Explain your answers.

P∞ n 9. Suppose n=1 an6 diverges. What can you say about P∞ n (a) the series n=1 an2 ? P∞ (b) the series n=1 an? P∞ n (c) the series n=1 an8 ? P∞ n (d) the series n=1 an(−3) ? P∞ n (e) the radius of convergence of the power series n=1 an(x − a) ? 10. Evaluate the following series or determine that they diverge: Review Chapter 11 (p 803), 27-31.

11. Determine whether the following series converge conditionally, converge absolutely, or diverge. Review Chapter 11 (p 803), 11,12,13,14,15,16.

12. Find the first 5 nonzero terms of the Taylor series of f(x) = sin x at a = π/6. State the linear

approximation of f about a = π/6. State the Taylor polynomial of degree 2 p2 for f about a = π/6. How large is the magnitude of the error in the approximation f(x) ≈ p2(x), for x ∈ [0, π/3], at most?

13. REVIEW Chapter 11 (p 804), # 47,48,50,51. (Find the Maclaurin series for f and its radius of convergence.)

14. Determine the radius and the interval of convergence of the following power series. ∞ X 3n(x + 4)n √ n n=1 15. REVIEW Chapter 11 (p 804), # 56 (approximate integral) Note: “correct to two decimals” means to within an error of 0.005.

16. (a) Write down a formula for the Taylor series of f(x) about a basepoint x = a.

(b) Write down a formula for the Taylor polynomial pn(x) of f(x) about x = a.

(c) If pn is used to approximate a function f about x = a,

f(x) ≈ pn(x)

Write down an upper bound for the error |f(x) − pn(x)|. x (d) Find p2(x) for f(x) = e about x = 0. Write down a formula for an upper bound for |f(x) − p2(x)|. Can you find an upper bound for |f(x) − p2(x)| if |x| < .1?