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.

Sum of an Infinite Geometric Series (80)

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:

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).

(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

Ratio Test (Convergence) (81) Does converge?

Ratio Test (Convergence) (81) Does converge?

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

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).

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

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

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

Partial Sums & Taylor Series (83)

(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:

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

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

Lagrange Form of the Remainder (84)

Lagrange Form of the Remainder (84)

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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.

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)

Parametric Acceleration (90) Let c (t) = (t2 + 1, t3 − 4t). Find the acceleration

Arc Length on a Function (91)

Arc Length on a Function (91)

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Arc Length on a Parametric Curve (92)

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

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

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.

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,

Shell Method (95)

Shell Method (95)

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

Shell Method (95) Shells

Simpson’s Rule (96) x –2 –1 1 2 y 6 3

Dot Product (97) (98) THEOREM 3 Product Rule for Dot Product Assume that r1(t) and r2(t) are differentiable. Then