Third Examination. Thursday, November 19, 1998.

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Problem1. % latex2html id marker 286 \( (\alph{enumi}) \) Find the solution of the differential equation \( {\displaystyle \frac{dy}{dx}=xy^{2}} \) which satisfies the initial condition \( y(0)=2 \).






















Problem2. % latex2html id marker 289 \( (\alph{enumi}) \) The differential equation \( {\displaystyle \frac{dP}{dt}=k(1-P)} \), where \( k \) is a positive constant, models performance \( P(t) \) of a trainee learning a skill. Performance is measured on a scale of 0 (no skill) to 1 (perfect performance), and \( t \) is elapsed time in hours since the beginning of training. Use the method of separation of variables to solve this differential equation for \( P(t) \), assuming the initial condition \( P(0)=1/3 \).
Problem3. % latex2html id marker 296 \( (\alph{enumi}) \) Determine whether the given sequence converges or diverges. If it converges, find the limit. Show all steps leading to your solution.
1.
\( {\displaystyle a_{n}= \frac{2n^{2}-3n+1}{3n^{2}+2n-1}} \)














2.
\( {\displaystyle b_{n}= \frac{1+n^{2}}{1+n}} \)














Problem4. % latex2html id marker 299 \( (\alph{enumi}) \) Determine whether the given series is convergent or divergent. If it is convergent, find the exact value of its sum. Show all work.
1.
\( {\displaystyle \sum_{n=1}^{\infty} \frac{8}{9^{n-1}}=8+\frac{8}{9}+ \frac{8}{9^{2}}+\cdots} \)














2.
\( {\displaystyle \sum_{n=1}^{\infty} \frac{(-5)^{n-1}}{4^{n}}=\frac{1}{4}-\frac{5}{4^{2}}+ \frac{5^{2}}{4^{3}}-\cdots} \)
Problem5. % latex2html id marker 302 \( (\alph{enumi}) \) Use the integral test to determine which of the following series is convergent and which is divergent. Show all work.
1.
\( {\displaystyle \sum_{n=1}^{\infty} \frac{n}{n^{2}+1}} \)














2.
\( {\displaystyle \sum_{n=2}^{\infty} \frac{1}{n\left(\ln(n)\right)^{2}}} \)














Problem6. % latex2html id marker 305 \( (\alph{enumi}) \) Test the series for convergence or divergence. In each case, name the test you used: divergence test; alternating series test; integral test; comparison test; limit comparison test; ratio test.
1.
\( {\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{1+\sqrt{n}}= \frac{1}{2}-\frac{1}{1+\sqrt{2}}+\frac{1}{1+\sqrt{3}}-\cdots} \)














2.
\( {\displaystyle \sum_{n=1}^{\infty} \frac{(-n)^{n+1}}{1+3n}= \frac{1}{4}-\frac{8}{7}+\frac{81}{10}-\cdots} \)
Problem7. % latex2html id marker 308 \( (\alph{enumi}) \) Determine whether the given series is absolutely convergent. In each case, name the test you used: divergence test; alternating series test; integral test; comparison test; limit comparison test; ratio test.
1.
\( {\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n+1}} \)














2.
\( {\displaystyle \sum_{n=1}^{\infty} \frac{n}{2^{n}}} \)