Study Guide for Test 4 of Mth 169

STUDY GUIDE FOR TEST 4 OF MTH 169

Test 4 is scheduled for Thursday, April 19. It will cover sections 10.6–9 and 11.1,2

Be familiar with the following terms 1. alternating series 2. absolute convergence 3. conditional convergence 4. power series 5. radius of convergence 6. interval of convergence 7. Taylor series 8. Maclaurin series 9. Taylor polynomial 10. remainder term of a Taylor series 11. parametric curve 12. parametric equations Know the following tests and theorems.

1. Alternating Series Test: If ⟨an⟩ is a sequence such that

(a) for all n, an ≥ 0

(b) for all n, an+1 ≤ an

2. Alternating Series Estimation Theorem: If ⟨an⟩ satisﬁes the conditions of the Alternating Series Test ∞ ∑ ∑n − k − − k ≤ then ( 1) ak ( 1) ak an+1 for all n. k=0 k=0 ∑∞ n 3. Convergence of Power Series: For every cn(x − a) one of the following is true. n=0 ∑∞ n (a) cn(x − a) converges absolutely for all x. n=0 ∑∞ n (b) There is R > 0 such that cn(x − a) converges absolutely for all x with |x − a| < R and diverges for n=0 all x with |x − a| > R. ∑∞ n (c) cn(x − a) converges if x = a and diverges if x ≠ a. n=0 ∑∞ ∑∞ n n 4. Multiplication of Power Series: If bn(x − a) and cn(x − a) both converge absolutely for all x n=0 n=0 with |x − a| < R then

[ ][ ] ∑∞ ∑∞ n n 1 2 bn(x − a) cn(x − a) = b0c0 + (b0c1 + b1c0)(x − a) + (b0c2 + b1c1 + b2c0)(x − a) + ··· n=0 n=0

converges absolutely for all x with |x − a| < R. ∑∞ ∑∞ n n 5. Substitution in Power Series: If cnx converges absolutely for all x with |x| < R and f(x) = cnx n=0 n=0 ∑∞ n then f(g(x)) = cn(g(x)) converges absolutely for all x with |x| < R. n=0 ∑∞ ∑∞ n n 6. Diﬀerentiation of Power Series: Let f(x) = cn(x − a) . If cn(x − a) converges absolutely for all n=0 n=0 x with |x − a| < R then f has derivatives of all orders and ∑∞ ′ n−1 f (x) = ncn(x − a) n=1 ∑∞ ′′ n−2 f (x) = n(n − 1)cn(x − a) n=2 . . ∞ ∑ n! f (k)(x) = c (x − a)n−k (n − k)! n n=k . .

∑∞ ∑∞ n n 7. Integration of Power Series: Let f(x) = cn(x − a) . If cn(x − a) converges absolutely for all x n=0 n=0 ∞ ∞ ∫ ∑ c ∑ c with |x − a| < R then f dx = n (x − a)n+1 + c and n (x − a)n+1 converges absolutely for all n + 1 n + 1 n=0 n=0 x with |x − a| < R. ∞ ∑ f (n)(a) 8. Taylor Series: f(x) = (x − a)n n! n=0 ∞ ∑ f (n)(o) 9. Maclaurin Series: f(x) = xn n! n=0

∑n f (k)(a) 10. Taylor Polynomial of Order n: P (x) = (x − a)k n k! k=0 f (n+1)(c) 11. Remainder of a Taylor Polynomial of Order n: R (x) = (x − a)n+1, where c is a number n (n + 1)! between x and a.

12. Taylor’s Formula: If f has derivatives of all orders on an open interval I containing a then for every x ∈ I and for every n, f(x) = Pn(x) + Rn(x).

∑∞ (n) f (a) n 13. Convergence of Taylor Series: (x − a) converges to f if and only if lim Rn(x) = 0. n! n→∞ n=0 dy dy 14. Derivative of a Parametric Curve: = dt dx dx dt 15. Area Bounded∫ by a Parametric Curve: The area between a parametric curve∫ C and the x-axis for ≤ ≤ b ′ ≤ ≤ b ′ a t b is a y(t)x (t) dt, and the area between C and the y-axis for a t b is a x(t)y (t) dt. ∫ √ b ′ 2 ′ 2 16. Length of a Parametric Curve: a (x ) + (y ) dt 17. Area of a Surface Generated by a Parametric Curve:∫ The√ area of the surface generated by rotating a parametric curve C about the x-axis for a ≤ t ≤ b is 2π b y(t) (x′)2 + (y′)2 dt, and the area of the surface ∫ √ a b ′ 2 ′ 2 generated by rotating C about the y-axis is 2π a x(t) (x ) + (y ) dt.

Useful Taylor Series ∞ ∑ xn 1. ex = , converges absolutely for all x n! n=0 ∞ 1 ∑ 2. = xn, converges absolutely for |x| < 1 1 − x n=0 ∞ ∑ (−1)n 3. sin x = x2n+1, converges absolutely for all x (2n + 1)! n=0 ∞ ∑ (−1)n 4. cos x = x2n, converges absolutely for all x (2n)! n=0 ∞ ∑ (−1)n+1 5. ln(1 + x) = xn, converges absolutely for |x| < 1 n n=1 xn 6. lim = 0 for all x n→∞ n!

The following is a list of things you should know and be able to do for the test. It is not necessarily complete. 1. Determine whether an alternating series converges. 2. Find a bound for the diﬀerence between the sum of an alternating series and the value of its partial sum. 3. Determine the number of terms needed in a partial sum of an alternating series so that the value of the partial sum is within a given distance of the sum of the series. 4. Determine whether a series converges absolutely, converges conditionally, or diverges. 5. Find the radius and interval of convergence of a power series. 6. Generate a power series representation of a function by multiplying two known power series. 7. Generate a power series representation of a function by substituting an expression in place of x in a known power series. 8. Diﬀerentiate a power series. 9. Integrate a power series. 10. Find a Taylor or Maclaurin series for a function and determine its radius of convergence.

11. Find Pn(x).

12. Find Rn(x). 13. Determine whether a Taylor series converges to the function which generated it. 14. Find the Taylor series for a function using known Taylor series and properties of power series. 15. Parametrize a curve. 16. Determine a relation between x and y which characterizes a given parametric curve. 17. Parametrize circular motion. ( ) dy 18. Find the slope of a line tangent to a parametric curve . dx 19. Find the area bounded by a parametric curve. 20. Find the length of a parametric curve. 21. Find the area of a surface obtained by rotating a parametric curve.