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**Sum of an Infinite Geometric Series (80)**

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

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**Sum of an Infinite Geometric Series (80)**

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**Sum of an Infinite Geometric Series (80)**

Write S as a sum of two geometric series. This is valid by Theorem 1 because both geometric series converge:

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**Ratio Test (Convergence) (81)**

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).

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**(81) Prove that converges.**

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

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**Ratio Test (Convergence) (81)**

Does converge?

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**Ratio Test (Convergence) (81)**

Does converge?

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**Ratio Test (Interval of Convergence) (81)**

Using the Ratio Test Where does converge? Step 1. Find the radius of convergence. Let and compute the ratio ρ of the Ratio Test: We find that Ratio Test

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**Ratio Test (Interval of Convergence) (81)**

Using the Ratio Test Where does converge? 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).

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**Maximum Error in the Approximation of an Alternating Series**

(82)

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**Maximum Error in the Approximation of an Alternating Series**

(82) Alternating Harmonic Series Show that converges conditionally. Then: (a) Show that (b) Find an N such that SN 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

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**Maximum Error in the Approximation of an Alternating Series**

(82) Alternating Harmonic Series Show that converges conditionally. Then: (a) Show that (b) Find an N such that SN 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

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**Partial Sums & Taylor Series (83)**

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**(83) We define the nth Taylor polynomial centered at x = a as follows:**

Taylor Series of f (x) centered at x = c: THEOREM 1 Taylor Series Expansion If f (x) is represented by a power series centered at c in an interval |x − c| < R with R > 0, then that power series is the Taylor series In the special case c = 0, T (x) is also called the Maclaurin series:

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

(83)

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**Partial Sums & Taylor Series (83)**

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

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**Lagrange Form of the Remainder (84)**

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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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(88)

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**Derivative of a Parametric Curve (89)**

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.

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**Derivative of a Parametric Curve (89)**

Let c (t) = (t2 + 1, t3 − 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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**Parametric Acceleration (90)**

Let c (t) = (t2 + 1, t3 − 4t). Find the acceleration

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**Arc Length on a Function (91)**

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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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**Slope of a Polar Curve (93)**

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 Since x = r cos θ and y = r sin θ, we use the chain rule.

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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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Shell Method (95)

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Shell Method (95) Implies everything is in terms of x.

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Shell Method (95) Shells

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Simpson’s Rule (96) x –2 –1 1 2 y 6 3

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**Dot Product (97) (98) THEOREM 3 Product Rule for Dot Product**

Assume that r1(t) and r2(t) are differentiable. Then

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Observe that x (t) = t 2 + 1 is an even function and that y (t) = t 3 − 4t is an odd function. As noted before Example 5, this tells us that c (t) is symmetric.

Observe that x (t) = t 2 + 1 is an even function and that y (t) = t 3 − 4t is an odd function. As noted before Example 5, this tells us that c (t) is symmetric.

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