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THEOREM 2 Sum of a Geometric Series Let c 0. If |r| < 1, then If |r| ≥ 1, then the geometric series diverges. Sum of an Infinite Geometric Series (80)

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Write S as a sum of two geometric series. This is valid by Theorem 1 because both geometric series converge: Sum of an Infinite Geometric Series (80)

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Does the given series converge? THEOREM 1 Ratio Test Assume that the following limit exists: (i) If ρ < 1, then converges absolutely. (ii) If ρ > 1, then diverges. (iii) If ρ = 1, the test is inconclusive (the series may converge or diverge). Ratio Test (Convergence) (81)

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Prove thatconverges. THEOREM 1 Ratio Test Assume that the following limit exists: (i) If ρ < 1, then converges absolutely. (ii) If ρ > 1, then diverges. (iii) If ρ = 1, the test is inconclusive (the series may converge or diverge). Compute the ratio and its limit with Note that (81)

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Does converge? Ratio Test (Convergence) (81)

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Does converge? Ratio Test (Convergence) (81)

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Using the Ratio Test Where doesconverge? Step 1. Find the radius of convergence. Let and compute the ratio ρ of the Ratio Test: We find that Ratio Test Ratio Test (Interval of Convergence) (81)

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Using the Ratio Test Where doesconverge? Step 2. Check the endpoints. The Ratio Test is inconclusive for x = ±2, so we must check these cases directly: Both series diverge. We conclude that F(x) converges only for |x| < 2 (Figure 2).Figure 2 Ratio Test (Interval of Convergence) (81)

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Maximum Error in the Approximation of an Alternating Series (82)

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Alternating Harmonic Series Show that converges conditionally. Then: (a) Show that (b) Find an N such that S N approximates S with an error less than 10 −3. The terms are positive and decreasing, and Therefore, S converges by the Leibniz Test. The harmonic series diverges, so S converges conditionally but not absolutely. Maximum Error in the Approximation of an Alternating Series (82)

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Alternating Harmonic Series Show that converges conditionally. Then: (a) Show that (b) Find an N such that S N approximates S with an error less than 10 −3. Now, applying Eq. (2), we have We can make the error less than 10 −3 by choosing N so that Maximum Error in the Approximation of an Alternating Series (82)

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Partial Sums & Taylor Series (83)

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THEOREM 1 Taylor Series Expansion If f (x) is represented by a power series centered at c in an interval |x − c| 0, then that power series is the Taylor series In the special case c = 0, T (x) is also called the Maclaurin series: We define the nth Taylor polynomial centered at x = a as follows: Taylor Series of f (x) centered at x = c: (83)

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Find the Taylor series for f (x) = x −3 centered at c = 1. (83)

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we can write the coefficients of the Taylor series as: The Taylor series for f (x) = x −3 centered at c = 1 is Partial Sums & Taylor Series (83)

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Lagrange Form of the Remainder (84)

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(85) (86) (87)

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(88)

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THEOREM 2 Slope of the Tangent Line Let c (t) = (x (t), y (t)), where x (t) and y (t) are differentiable. Assume that CAUTION Do not confuse dy/dx with the derivatives dx/dt and dy/dt, which are derivatives with respect to the parameter t. Only dy/dx is the slope of the tangent line. Derivative of a Parametric Curve (89)

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Let c (t) = (t 2 + 1, t 3 − 4t). Find: (a) An equation of the tangent line at t = 3 (b) The points where the tangent is horizontal. Derivative of a Parametric Curve (89)

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Let c (t) = (t 2 + 1, t 3 − 4t). Find the acceleration Parametric Acceleration (90)

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Arc Length on a Function (91)

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(91)

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Arc Length on a Parametric Curve (92)

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Calculate the length s of one arch of the cycloid generated by a circle of radius R = 2. (92)

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Calculate the length s of one arch of the cycloid generated by a circle of radius R = 2. (92)

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Since x = r cos θ and y = r sin θ, we use the chain rule. To find the slope of a polar curve r = f (θ), remember that the curve is in the x-y plane, and so the slope is Slope of a Polar Curve (93)

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Horizontal & Vertical Tangents of a Polar Curve (93) (94) The equation r = 4 sin θ defines a circle of radius 2 tangent to the x-axis at the origin. Find its horizontal and vertical lines,

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Shell Method (95)

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Implies everything is in terms of x.

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Shell Method (95)

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Simpson’s Rule (96) x –2–2 –1–1 012 y

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Dot Product (97) (98) THEOREM 3 Product Rule for Dot Product Assume that r 1 (t) and r 2 (t) are differentiable. Then

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