Key, Exam 2, Mat222, Spring 2004

The student starts off well, but gets into inequality trouble! Even so, s/he is on the right track: use integrals to bound the remainder, and use the one at the upper limit to provide a (reasonable!) upper bound.
I ask this student two things:
- whether 999 terms are actually necessary, and
- to compare with the correct answer (ln(2)).
The second question might shed some light on the former....
This is a pretty good collection of ideas (the dots were never quite connected, however!): first the student bounded the given integral by another integral (the larger's convergence would force the other to converge). It's never stated, but their integrands are both positive on the interval of integration.
Then the student considers related series, and makes some nice arguments at the end: if c converges, then so does b; if b converges, then so does a. In fact the rightmost improper integral converges if and only if the related series c converges (see The Integral Test, p. 749), which leads to the answer of the question asked!

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