M408D First Midterm, October 3, 2002


1. A sequence { tex2html_wrap_inline36 } is said to ``grow faster'' than { tex2html_wrap_inline38 } if tex2html_wrap_inline40 . Put the following sequences in order of growth rate, from fastest to slowest. JUSTIFY YOUR ANSWERS.






2. Evaluate the following limits or improper integrals, or write DNE if the limits do not exist.

a) tex2html_wrap_inline52

b) tex2html_wrap_inline54

c) tex2html_wrap_inline56

d) tex2html_wrap_inline58

3. Which of the following series converge, and which diverge. In each case, give a 1-sentence explanation (e.g., ``converges by ratio test'', or ``diverges by comparison to tex2html_wrap_inline60 '')

a) tex2html_wrap_inline62

b) tex2html_wrap_inline64

c) tex2html_wrap_inline66

d) tex2html_wrap_inline68

4. a) Write down the Taylor series for tex2html_wrap_inline70 .

b) Write down the 6-th order Taylor polynomial (i.e., through the tex2html_wrap_inline72 term) for tex2html_wrap_inline74 .

c) Evaluate tex2html_wrap_inline76 . (That is, the 6-th derivative of f(x), evaluated at x=0.)

d) Evaluate tex2html_wrap_inline82 to five decimal places. (No, you don't need a calculator!)

5. a) Find the second-order Taylor polynomial for tex2html_wrap_inline84 around x=4.

b) What is the radius of convergence of the power series tex2html_wrap_inline88 ? Justify your answer.

Lorenzo Sadun

Thu Oct 3 09:57:41 CDT 2002