Find the sum of the finite geometric series find the sum of

Question

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.