Math 143 Flash Cards. Divergence of a sequence {a n } {a n } diverges to. Sandwich Theorem for Sequences. Continuous Function Theorem for Sequences

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1 Math Flash cards Math 143 Flash Cards Convergence of a sequence {a n } Divergence of a sequence {a n } {a n } diverges to Theorem (10.1) {a n } diverges to Sandwich Theorem for Sequences Theorem (10.1) Continuous Function Theorem for Sequences sequence bounded above sequence bounded below lower bound of a sequence

2 lim a n exists and is a number. n I hope that they are useful! lim a n = n lim a n does not exist (including = or = ). n lim a n = n

3 upper bound of a sequence greatest lower bound of a sequence least upper bound of a sequence sequence being bounded sequence being unbounded sequence being nondecreasing Theorem (10.1) sequence being nonincreasing Monotonic Sequence Theorem Definition (10.2) Definition (10.2) infinite series n-th term of a series k=1 a k

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5 Definition (10.2) Definition (10.2) n-th partial sum of a series k=1 a k a k converges k=1 Definition (10.2) Definition (10.2) a k converges to L k=1 a k diverges k=1 Definition (10.2) Definition (10.2) Geometric Series When a geometric series converges? to what? Theorem (10.2) Theorem (10.2) The n-th term test for divergence When is the n-th term test for divergence inconclusive? Definition (10.2) Theorem (10.3) a k converges to L k=1 State the integral test

6 When lim a n equals ZERO, the n-th term test n for divergence is inconclusive for a n. (It does not tell you anything.)

7 Theorem (10.3) Meaning (10.3) What three things have to mentioned or checked when using the Integral Test? What does it mean for a series to be positive? Meaning (10.3) Meaning (10.3) What does it mean for an series to be continuous? What does it mean for an series to be decreasing? Match It! (10.3) Yes or No! (10.3) When does a p-series converge? Does 1 n converge? Yes or No! (10.3) Yes or No! (10.3) Does 1 3 n converge? Does 1 n 8 converge? Theorem (10.4) Theorem (10.4) State the Comparison Tests If you expect a n to converge and what to use the Comparison Test, do you use d n a n or a n c n?

8 1. A function is positive. 2. A function is continuous. This does not mean anything!!! 3. A function is descreasing. This does not mean anything!!! This does not mean anything!!! p > 1 No. It is the harmonic series. Yes (p = 8 > 1) No (p = 1/3 < 1) a n c n

9 Theorem (10.4) Theorem (10.4) If you expect a n to diverge and what to use the Comparison Test, do you use d n a n or a n c n? State the Limit Comparison Test. Theorem (10.4) Theorem (10.5) When is the Limit Comparison Test inconclusive? State the Ratio Test Theorem (10.5) Theorem (10.5) When is the Ratio Test inconclusive? State the Root Test Theorem (10.5) Definition (10.6) When is the Root Test inconclusive? State the definition of an alternating series? Theorem (10.6) Definition (10.6) State Leibnitz s test for alternating series State the definition of absolutely convergent series

10 d n a n

11 Definition (10.6) Practice (10.6) State the definition of conditionally convergent series State four examples of absolutely convergent series Practice (10.6) Practice (10.6) State four examples of conditionally convergent series State four examples of divergent series Theorem (10.6) Theorem (10.6) State the absolute convergence test State the Rearragement Theorem for Absolutely Convergent Series Definition (10.7) Definition (10.7) Definition of power series about x = a Center of a power series Technique (10.7) Definition (10.7) How to find the center of a power series? Definition of radius of converence of a power series

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13 Technique (10.7) Practice (10.7) How to find the radius of converence of a power series? State the power series about x = 0 which 1 equals 1 x. Technique (10.7) Practice (10.7) How do we differentiate a n x n n=5 How do we differentiate a n x 5n n=3 Technique (10.7) Practice (10.7) How do we integrate a n x n How do integrate a n x 5n? Practice (10.7) Practice (10.7) What is wrong with x n dx = x n+1 n + 1? What is wrong with x n dx = C + xn+1 n + 1? Practice (10.7) Definition (10.8) What is wrong with x n dx = C + x n+1 n + 1? Taylor series generated by f at x = a

14 x n = 1 + x + x 2 + x x n + n=1 d dx a n x 5n = n=3 n=3 a n d dx x5n = n=3 5 n a n x 5n 1 d dx a n x n = n=5 n=5 a n d dx xn = n a n x n 1 n=5 an x 5n dx = a n x 5n dx = C+ a n x 5n+1 n + 1 an x n dx = a n x n dx = a n x n+1 n+1 You are adding an infinite number of C s all of which are equal. C diverges for any nonzero number C. n=1 No +C. Nothing!

15 Definition (10.8) Definition (10.8) MacLaurin series generated by f(x) Taylor Polynomial of order n generated by f Theorem (10.9) Theorem (10.9) State Taylor s Theorem State Taylor s Formula Theorem (10.9) Theorem (11.2) State the Remainder Estimation Theorem Paramterized Curve being differentiable Formula (11.2) Formula (11.2) dy dx for paramteric curve x = f(t), y = g(t) d 2 y for paramteric curve x = f(t), dx2 y = g(t) Formula (11.2) Formula (11.2) Arc length for paramteric curve x = f(t), y = g(t), a x b Area of surface of revolution when revolving around x-axis and y 0

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17 Formula (11.2) Technique (11.4) Area of surface of revolution when revolving around y-axis and x 0 For a polar curve, how to compute dy dx when r = 0? Formula (11.5) Formula (11.5) For a polar curve, compute the arc length. For a polar curve, compute the area bounded (simple). Formula (11.5) Definition (12.1) For a polar curve, compute the area bounded (more complicated). Definition of sphere Formula (stuff) Match It! (stuff) a b =?? textbf a b cos(θ) =?? Formula (stuff) Formula (stuff) textbfa b =?? textbf a b sin(θ) =??

18 The sphere with center (a, b, c) and radius r is the set of points whose distance from (a, b, c) is exactly equal to r. textbfa b cos(θ) = a b a b = textbfa b cos(θ) textbfa b sin(θ) = textbfa b textbfa b = textbfa b sin(θ)

19 Meaning? (stuff) Meaning? (stuff) a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 r = x 0 + td 1, y 0 + td 2, z 0 + td 3 Meaning? (stuff) d (r dt 1(t) r 2 (t))

20 r = x 0 + td 1, y 0 + td 2, z 0 + td 3 is the vectorequation of a line which contains the point (x 0, y 0, z 0 ) and has direction vector d 1, d 2, d 3. The equation a(x x 0 )+b(y y 0 )+c(z z 0 ) = 0 is the equation of a plane which contains the point (x 0, y 0, z 0 ) and normal vector n = a, b, c. d dt (r 1(t) r 2 (t)) = r 1(t) r 2 (t)r 1 (t) r 2(t)