Part 1: The Infinite Geometric Series 1. Given the following geometric series wr

Question

Part 1: The Infinite Geometric Series 1. Given the following geometric series write out the first 5 partial sums. Explain how to do this. %u2211_(n=1)^%u221E%u2592%u30164(1/3)^n %u3017 S_1 = S_2 = S_3 = S_4 = S_5

2. Do these partial sums appear to be approaching a finite sum for the entire series? If so, what does that sum appear to be?

3. How do we know for sure that the above infinite geometric series has a sum? What is the formula that we can use to find the sum of an infinite geometric series?

4. Identify the following and use them to determine the sum of the infinite geometric series above. r = a_1= Sum =

5. Can we find the sum of these infinite geometric series? In other words, do these series converge? Explain. If you can find the sum, do so. a) %u2211_(n=1)^%u221E%u2592%u30163(-5/2)^n %u3017 b) %u2211_(n=1)^%u221E%u2592%u30161/4 (.125)^n %u3017

Part II: Maclaurin Series Directions: Determine the Maclaurin Series for the following function: f(x)=e^(-3x) for n = 4

1. Determine the first four derivatives of f(x)=e^(-3x). f^I (x)= f^II (x)= f^III (x)= f^IV (x)=

2. Evaluate f(x)=e^(-3x) and the derivatives above at x = 0 f(0)= f^I (0)= f^II (0)= f^III (0)= f^IV (0)=

3. The formula for the Maclaurin Series is f(0)+f^I (0)x+(f^II (0))/2! x^2+f^III/3! x^3+%u22EF+(f^n (x))/n! x^n. Plug the values from question 2 into the formula. Then, simplify the fractions and write the Maclaurin Series approximation for f(x)=e^(-3x).

4. Enter f(x)=e^(-3x) and the series approximation from step 3 into a graphing calculator. View the graphs and the table of values. Does the Maclaurin series approximation appear to converge on f(x)=e^(-3x) for all values of x? If so, what does this mean about when I can use the Maclaurin series to approximate f(x)=e^(-3x)? If not, for what values of x can I use the series to approximate f(x)?

5. What would happen to the accuracy of the approximation the Maclaurin Series provides if I increased the number of terms in the series? Why?

6. Determine a Maclaurin Series approximation for f(x)=sin%u2061(6x) where n = 6.

Part II: Using MacLaurin Series Maclaurin Series expansions

1. Use one of the expansions above to write the Maclaurin Series for cos%u2061(2x^2). (Hint: What should you replace x with in the expansion you choose?

2. Use one of the expansions above to write the Maclaurin Series for (1+5x)^3.

3. Use one or more of the expansions above to write the first 3 terms of the Maclaurin Series for xe^(-x). Hint: Use the first three terms from the relevant series and multiply.

4. Use a Maclaurin series to approximate the value of the following: %u222B_0^2%u2592%u3016ln%u2061(x+1)dx%u3017.

5. Why is it important that we be able to use series like the Maclaurin to approximate the behavior of various functions? Under what circumstances would we prefer to use a series approximation to model the behavior of a function rather than the function itself?

Explanation / Answer

please post the questions again. Its not showing up properly.

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