AMT Homework 1 Supplement

Dr. Osborne 2012 The purpose of this supplement is to illustrate the techniques of expansions and some of the methods used to finesse them.

1. Obtain the Taylor expansion of the function 1

5 2x about x = 0, 2, and -3. Guess the radius of convergence of each expansion from the location of the pole, then determine the radius using the ratio test.

2. Determine the Taylor expansion of xe about x = 0 and 1. What is the radius of convergence of each

expansion? Are your answers consistent?

3. Determine the Maclaurin expansion of 221 xe

x

and its radius of convergence.

4. Determine the Maclaurin expansion of 3

1

2 x and its radius of convergence. Can you explain why the radius of convergence you obtained is the correct answer without using the ratio test?

5. Use the Maclaurin expansion of 3cos x to determine the exact value of 6 3120 1 2 cos 2limx x xx

.

Check your answer with a calculator using the so-called tabular method. Are they consistent? Would LHopitals rule work to determine this limit? Would it be convenient? Explain what the LHopital approach would require you to do, without actually using it.

6. Write the first 4 nonzero terms in the Maclaurin expansion of 325 2 3x . Your terms may include the irrational number 5 8 , but all other numbers must be rational. Make a table showing the exact value of the function and the value given by your expansion (use a calculator for this!!) for x = 0.1, 0.5, 0.7, and 1.0. What can you say about the validity of your expansion? Why should it behave in this way?

7. The analysis that lead to the integral test also provides a way for us to bound the value of sums: both

infinite and finite. In the following, use the given value of the finite sum to bound the value of the desired sum. You should obtain both a lower and an upper bound for the desired sum.

(a) Given 1000

21

11.075674547...

1n n , bound 21

1

1nS

n

. The exact sum can be shown to equal

( coth 1) 2 1.076674047... from contour integration. Is this value within your bounds?

(b) 1

1nn

kH

k . Find 1210H , the sum of the first trillion terms in the Harmonic series, given

that 1000 7.485470861...H . How accurate is your estimate (how far apart are your bounds)? About how many terms must be summed for the total to cross 100?

8. Bound the value of 1210

1

1

n n from above and below, given that 1000

1

161.80101...

n n . What can you

say about the accuracy of your results (how far apart are your bounds)?.