Merced College; Don Power

Math 4B -- Lab 6: Families of Series: Developing One Series From Another

Write out the first 6 terms for each of the 15 series you derive below.

In each term, leave products (especially factorials) in factored form (for example, use 2×3! instead of multiplying to get 12).

	Use the series for	To derive the series for	Required Operation:
Family 1
1.	e^x	e^-x	Replace (x) by (-x)
2.	e^x and e^-x	sinh(x)	Since sinh(x)=, you can subtract the series and divide the result by 2.
3.	sinh(x)	cosh(x)	You can differentiate or integrate term by term. If you integrate, you have to calculate the value of the integration constant C (by letting x=0)
4.	e^x	e^ix	Replace (x) by (ix). Clean up powers of i: i^2= -1
5.	e^x	e^-ix	What must you replace (x) by ? _____
6.	e^ix and e^-ix	cos(x)	You already know the series for cos(x). The goal here is to find a formula for cos(x) similar to the one in #2 for sinh(x): by adding or subtracting the series for e^ix and e^-ix and dividing the result by 2.
7.	e^ix and e^-ix	i×sin(x)	Add or subtract and divide by 2. Based on the result, derive a formula for sin(x) similar to the one in #2 for sinh(x).
Family 2
8.	(1+x)^k		Start with the binomial series Replace k by ______
9.			Replace x by _________ What is the resulting series called? __________________
10.			Replace x by ________
11.		arctan(x)	Should you differentiate or integrate? If you integrate, remember to calculate C
12.		ln(1+x)	Should you differentiate or integrate? If you integrate, remember to calculate C
13.	ln(1+x)	ln(x)	Replace the x in (1+x) by ______
14.	(1+x)^k		You need two simultaneous changes: Replace x by ________ Replace k by _________
15.		arcsin(x)	Should you differentiate or integrate? If you integrate, remember to calculate C

Return to: Merced College; Don Power Updated 11/9/07 by Don Power