The Taylor and Fourier Series are among the most important ideas in mathematics that you will encounter during this class.

1. Find the first 5 terms of the sequence: , n = 0,1,2,3,

2. Find the sum of the finite geometric series:

3. Give the first four terms of the infinite series for a_n =   . Write the infinite series in sigma notation.

4. Find the sum of the finite geometric series using your CAS SYSTEM:

5. Find the sum of the finite geometric series using your CAS SYSTEM:

6. Find the first six partial sums of the following series to determine if it converges

or diverges.

            S₁ = _______________________                       S₂ = _______________________

            S₃ = _______________________                       S₄ = _______________________

            S₅ = _______________________                       S₆ = _______________________

            Convergent or divergent?

7. Find the first six partial sums of the following series to determine if it converges or diverges.  

            S₀ = _______________________                       S₁ = _______________________

            S₂ = _______________________                       S₃ = _______________________

            S₄ = _______________________                       S₅ = _______________________

            Convergent or divergent?

^{Circle one.}

8. When writing a transcendental equation as a polynomial, when would a Taylor series be

a better choice than a Maclaurin series to approximate a function ?

9. Determine the Maclaurin Series for the following function: f(x) = e^2x for n = 4

10. Enter f(x) = e^2x and the series approximation from the problem above into your CAS SYSTEM.

Sketch the graphs. Approximate the values of x for which the Maclaurin series appear to converge on f(x) = e^2x      

            Interval of convergence = ______________________

11. Use your CAS SYSTEM to find a Maclaurin Series approximation for f(x) = x sin(2x) where n = 6.

Maclaurin Series expansions

For problems 12Ac€?o16, use the expansions above.

12. Use one of the expansions above to write the Maclaurin Series for sin(3x⁵).

13. Use one of the expansions above to write the Maclaurin Series for (3 + x)⁴

14. Use one or more of the expansions above to write the first 3 terms of the Maclaurin Series for x²e^2x.

15. Use four terms of a Maclaurin series to approximate the value of the following:

Show the -Maclaurin integral, the integral of the Maclaurin expansion of the function.

16. Calculate each value below using five terms of a Maclaurin series expansion. Then find the calculator value correct to 8 places.

   e^0.2

cos(3.57)   

ln(2.8)   

17. Graph ln(x), the Maclaurin series used in problem 16 and then the Taylor series found in problem 17. Change your viewing window (green diamond F2) to x_min = -.5,

x_max = 10, xscl =.5, y_min= -3, y_max = 5 , y_scl = 1. Then sketch and label each graph.

Which series is a closer fit to ln(x) ? Why? For what values of x can these series be used to approximate f(x)?     

18.) Use your calculator to determine a Taylor Series approximation for f(x) = cos (px) where

n = 4 and a = 3.

19. Find the Fourier series for the function f(x) = sin x , for 0 < x < 2p

            Give the values correct to four decimal places.

          a₀ = _________________________

            a₁ = _________________________

a₂ = _________________________

a₃ = _________________________

b₁ = _________________________

b₂ = _________________________

b₃ = _________________________

f(x) = ________________________________________________________________

_______________________________________________________________

_______________________________________________________________

20. Find at least three non-zero terms and at least two cosine and two sine terms of the Fourier series for    

Find each value correct to four decimal places.