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Unformatted text preview: Tufts University
Math 12 Department of Mathematics April 4, 2005
Exam 3 Instructions. No calculators, notes or books allowed. You must show your work in the blue
book and cross out any work you do not want graded. Remember to sign your blue book.
With your signature you are pledging that you have neither given nor received assistance
on this exam. Most of the problems have multiple parts and several of them can be done
without using the other parts. Good luck! Use the geometric series °° 1
n: , < 1 *
“Ezoﬁw 1 _, m < > to give your answer to Problems 1—4. 1. (14 points) [Use the geometric series above] 1 (a) (7 points) Find a power series representation, centered at a, = 0, for f = (b) (7 points) Find a power series representation for f : (1 +902. 2. (14 points) [Use the geometric series above] (a) (8 points) Find the function represented by an"_1 and the radius of conver—
11:1 gence of the series. (b) (6 points) Use your answer to part (a) to compute Z 372—” exactly. n=1 3. (12 points) Using the geometric series above, ﬁnd a power series representation for
f = a7 ln(1 — :r) and indicate the radius of convergence. 4. (18 points) [Use the geometric series above] da:
11503 (a) (8 points) Find a power series for / da: 1 + 3 satisﬁes the conditions of the
3: 1/10
(b) (4 points) Check that the series for /
0 Alternating Series Theorem. (c) (6 points) Use the alternating series estimate to ﬁnd a ﬁnite sum that approxi— 1/10 dx
mates / 1 + 3 with an error satisfying error < 10—6. Write out your ﬁnite
0 3: sum but do not simplify. Emam continues on next page 5. (19 points) Let f(:1:) : Z“: n:1 (a) (12 points) Find the radius R and interval I of convergence of the above power
series. (b) (7 points) Write the inﬁnite series for f’ and give its radius of convergence. 6. (6 points) If the Maclaurin series for = 2(3: — 1)2 + 3(x — 1) + 6 is Z cum", ﬁnd
n20 on for all n 2 0. 7. (17 points) This problem should be done on a new page in your blue book. Write only
the number of each part followed by the answer that ﬁlls in the blank. You do not
need to simplify coefﬁcients or factorials in this problem. (a) (3 points) The formula for a Taylor Series for a function f expanded about
a = 7r is The following table gives the ﬁrst four derivatives of the function f : fm) :
sin (50/2). Use the above table to help you ﬁll in the following blanks: (b) (3 points) For n = 2k; even, ﬁnd the general form of f(”)(7r) =
(c) (3 points) For n : 2/6 + 1 odd, ﬁnd the general form of f(”)(7r) : (d) (4 points) The Taylor polynomial T4(:c) of degree 4 expanded about a : 7r is
T4<$> 2 Recall that if T(:c) is the Taylor series for a function f about a, Tn(:c) de
notes its Taylor polynomial of degree n about a, and = — T7,,(:C),
then Taylor’s Inequality says: f(”+1)(:c) S M for :c — a S d implies that M S (n + 1)! x — al"“ for x — a S d. (e) (4 points) Taylor’s inequality, applied to f = sin yields for all CE, 0 3 a7 3 27% R2($) S ...
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 Spring '08
 GARANT
 Power Series, Taylor Series, Mathematical Series, blue book

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