**Unformatted text preview: **09/25/2000 MON 11:38 FAX 6434330 MOFFITT LIBRARY .001 P. Vojta Math lBM Second Midterm Thu 23 Mar 2000 00
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1. (12 points) Is the series Z 273,2 absolutely convergent, conditionally convergent, or n21
divergent? OO
1
(14 points) Describe how one can compute Z —§ to Within 0.00005.
TL
n21
(You do not need to actually carry out the computation, but if your answer involves, say,
the nth partial sum, then you should say What 71 is.) to l 3- (12 points) Is the series 2 (1—)2
n 1].” 11:2 absolutely convergent, conditionally convergent, or divergent? 00
4. (12 points) Is the series 2 an, where ”:1 n+1/ﬁ" —% if n is even 1 if n is odd, or
an : absolutely convergent, conditionally convergent, or divergent? Explain. [Fewer than 10% of the students even got so far as to approach the main difﬁculty of this
problem.] 5. (18 points) (a). Find the Taylor polynomial, T3 (:6), for f(:r:) = 506‘” (centered about a 2 0). (b). Use Taylor’s Inequality to ﬁnd an upper bound for the error in using your answer to
(a) to compute f(1). 6. (20 points) (a). Show that the series 00 a:
is a solution of the differential equation
3;, : l + my _
(b). Over What interval is it a solution? 7. (12 points) Find the curve through the point (1, 1) that is everywhere orthogonal to the
family of curves 3; = 0:133. Y ...

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