Question

1-(Partial Fraction Decomposition Revisited) Consider the rational function 1/(1-x)(1-2x)

(a) Find power series expansions separately for 1/(1 − x) and 1/(1 − 2x).

(b) Multiply these two power series expansions together to get a power series ex-pansion for

1 (1−x)(1−2x)

(This involves doing an infinite amount of distributing and combining coeffi-cients, but you should be able to figure out the pattern here.)

c) Separate the power series in terms of power series for A/(1 − x) and B/(1 − 2x) for some constants A and B. (This was definitely not easier, but it is interesting

to know there is another way.

2- Time to remember some things from Calculus I)

(a) If the function f has a local maximum at a, must the second Taylor polynomial for f at a also have a local maximum at a?
(b) If the second Taylor polynomial for f at a has a local maximum at a, must the function f have a local maximum at a?

(c) If the function f has an inflection point at a, what does the second Taylor polynomial for f at a look like?

Answer 1