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

2. 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

3. Use Pascal’s triangle to expand the binomial (2a + 4b)4.
A. 16a a a a + 256
B. 16a4 + 32a3b + 64a2b ab b4
C. 16a a3b + 384a2b ab b4
D. 16a4 + 32a3b + 24a2b2 + 8ab3 + b4
5–Minute Check 1

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

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

6. 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

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

8. 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

9. power series
exponential series
Euler’s Formula
Vocabulary

10. Key Concept 1

11. 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

12. 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

13. 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

14. 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

15. Power Series Representation of a Rational Function
Example 1

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

17. 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

18. 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

19. Key Concept 2

20. 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:
Example 2

21. 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 is reasonable. 
Example 2

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

23. Key Concept 2

24. 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

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

26. 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

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

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

29. Key Concept 4

30. Key Concept 4

31. 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– for a > 0
= Simplify.
Example 4

32. 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

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

34. 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
 i Use a calculator to compute ln 2.
Answer: i
Example 5

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

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