Functions as Infinite Series LESSON 10–6 Functions as Infinite Series

Five-Minute Check (over Lesson 10-5) Then/Now New Vocabulary Key Concept: Power Series Example 1: Power Series Representation of a Rational Function Key Concept: Exponential Series Example 2: Exponential Series Key Concept: Power Series for Cosine and Sine Example 3: Power Series for Cosine and Sine Key Concept: Euler’s Formula Key Concept: Exponential Form of a Complex Number Example 4: Write a Complex Number in Exponential Form Example 5: Natural Logarithm of a Negative Number Lesson Menu

Use Pascal’s triangle to expand the binomial (2a + 4b)4. A. 16a4 + 128a3 + 384a2 + 512a + 256 B. 16a4 + 32a3b + 64a2b2 + 128ab3 + 256b4 C. 16a4 + 128a3b + 384a2b2 + 512ab3 + 256b4 D. 16a4 + 32a3b + 24a2b2 + 8ab3 + b4 5–Minute Check 1

Use Pascal’s triangle to expand the binomial (2 – y)6. A. 64 – 192y + 240y 2 – 160y3 + 60y4 – 12y5 + y6 B. 64 + 192y + 240y 2 + 160y3 + 60y4 +12y5 + y6 C. 64 – 32y + 16y 2 – 8y3 + 4y4 – 2y5 + y6 D. 64y6 – 192y5 + 240y 4 – 160y3 + 60y 2 – 12y 5–Minute Check 2

Find the coefficient of the 5th term in the expansion of (a + b)9. 5–Minute Check 3

Find the coefficient of the x 2y 2 term in the expansion of (3x – y)4. B. 9 C. 36 D. 54 5–Minute Check 4

Which expression represents the expansion of (x – 5)4 using sigma notation? B. C. D. 5–Minute Check 5

Use a power series to represent a rational function. You found the nth term of an infinite series expressed using sigma notation. (Lesson 10-1) Use a power series to represent a rational function. Use power series representations to approximate values of transcendental functions. Then/Now

power series exponential series Euler’s Formula Vocabulary

Key Concept 1

Power Series Representation of a Rational Function Use to find a power series representation of Indicate the interval on which the series converges. Use a graphing calculator to graph g(x) and the sixth partial sum of its power series. Example 1

Power Series Representation of a Rational Function To find the transformation that relates f(x) to g(x), use u-substitution. Substitute u for x in f, equate the two functions, and solve for u as shown. g(x) = f(u) Example 1

Power Series Representation of a Rational Function Therefore, g(x) = f(x – 3). Replacing x with x – 3 in f(x) = for |x | < 1 yields for |x – 3| < 1. Therefore, g(x) = can be represented by the power series . This series converges for |x – 3| < 1, which is equivalent to –1 < x – 3 < 1 or 2 < x < 4. Example 1

Power Series Representation of a Rational Function The sixth partial sum S6(x) of this series is or 1 + (x – 3) + (x – 3)2 + (x – 3)3 + (x – 3)4 + (x – 3)5 . The graphs of g(x) = and S6(x) = 1 + (x – 3) + (x – 3)2 + (x – 3)3 + (x – 3)4 + (x – 3)5 are shown. Notice that on the interval (2, 4), the graph of S6(x) comes close to the graph of g(x). Example 1

Power Series Representation of a Rational Function Example 1

Answer: g(x) = for 2 < x < 4 Power Series Representation of a Rational Function Answer: g(x) = for 2 < x < 4 Example 1

Use to find a power series representation of Use to find a power series representation of . Indicate the interval on which the series converges. Use a graphing calculator to graph g(x) and the sixth partial sum of its power series. Example 1

A. g(x) = for –2 < x < 2; B. g(x) = for –1 < x < 1; C. g(x) = for < x < ; D. g(x) = for –3 < x < 3; Example 1

Key Concept 2

Exponential Series Use the fifth partial sum of the exponential series to approximate the value of e1.25. Round to three decimal places. x = 1.25 Simplify. Answer: 3.458 Example 2

Exponential Series CHECK A calculator, using a partial sum of the exponential series with many more terms, returns an approximation of 3.49 for e1.25.Therefore, an approximation of 3.458 is reasonable.  Example 2

Use the fifth partial sum of the exponential series to approximate the value of e0.75. Round to three decimal places. A. 0.472 B. 2.102 C. 2.115 D. 2.117 Example 2

Key Concept 2

Power Series for Cosine and Sine A. Use the fifth partial sum of the power series for cosine to approximate the value of cos . Round to three decimal places. Example 3

Power Series for Cosine and Sine Simplify. Answer: 0.707 CHECK A calculator, using a partial sum of the power series for cosine with many more terms, returns an approximation of 0.707 for cos .Therefore, an approximation of 0.707 is reasonable.  Example 3

Power Series for Cosine and Sine B. Use the fifth partial sum of the power series for sine to approximate the value of sin . Round to three decimal places. Example 3

Power Series for Cosine and Sine Simplify. Answer: 0.259 CHECK A calculator, using a partial sum of the power series for sine with many more terms, returns an approximation of 0.259 for sin . Therefore, an approximation of 0.259 is reasonable.  Example 3

Use the fifth partial sum of the power series for sine to approximate the value of sin . Round to three decimal places. A. 0.207 B. 0.208 C. 0.210 D. 0.998 Example 3

Key Concept 4

Key Concept 4

Write 1 + i in exponential form. Write a Complex Number in Exponential Form Write 1 + i in exponential form. Write the polar form of 1 + i. In this expression, a = 1, b = 1, and a > 0. Find r. Simplify. Now find .  = tan–1 θ = tan–1 for a > 0 = Simplify. Example 4

Therefore, because a + bi = rei, the exponential form of 1 + i is . Write a Complex Number in Exponential Form Therefore, because a + bi = rei, the exponential form of 1 + i is . Answer: Example 4

Write 2 + 2i in exponential form. B. C. D. Example 4

ln (–2) = ln 2 + i ln (–k) = ln k + i Natural Logarithm of a Negative Number Find the value of ln (–2) in the complex number system. ln (–2) = ln 2 + i ln (–k) = ln k + i  0.693 + i Use a calculator to compute ln 2. Answer: 0.693 + i Example 5

Find the value of ln (–7) in the complex number system. A. 1.946 + i B. 1.946 C. 2.079 + i D. 1.946 +  Example 5

Functions as Infinite Series LESSON 10–6 Functions as Infinite Series