SampleSeries1Solns - SAMPLE PROBLEMS In each of the following problems decide if the series converges or diverges and specify which test(s you are

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Unformatted text preview: SAMPLE PROBLEMS In each of the following problems decide if the series converges or diverges and specify which test(s) you are using to make this conclusion. 1. 7. n=1 tan-1 n 1 + n2 Converges: since 0 < tan-1 n < , use Com2 1 parison with n2 . Then use Integral test or pseries test. 8. k=1 (-1)n-1 3-n n=1 Converges Absolutely: Ratio test 2. n=1 1 (k + 1)(k + 2) n2 (n + 1)n 1 Convergent: use Limit Comparison with . k2 Then use Integral test or p-series test. This is a telescoping series, so use partial fractions and find actual sum. Converges: Ratio test 9. 3. n=1 1 n - ln(n) 1 n. n=1 1 + 2n 1 + 5n n Diverges: Limit Comparison with use Integral test or p-series test. 4. k=1 Then 2 Convergent: use Limit Comparison with 5n . Then use Ratio test, Root test or Geometric series. 10. k 3+1 k 1 n2 . k=1 2 + sin k k2 Converges: Limit Comparison with use Integral test or p-series test. 5. n=1 Convergent: since 1 2+sin k 1, use Com1 Then parison with k 2 . Then use Integral test or pseries test. 11. 5 2 -n k=1 sin k Converges: Use Ratio test, Root test or Geometric series. 6. n=1 Divergent: since limx0 sin x = 1, useLimit x 1 Comparison with k . Then use Integral test or p-series test. 12. 1 k2 + k 1 n. ne-n n=1 Diverges: Limit Comparison with use Integral test or p-series test. Then Convergent: use Ratio test. 1 13. k=3 3 k(ln k)2 Convergent: use Integral test with f (x) = 1 . x(ln x)2 14. n=1 n! (n + 2)! Convergent: use Ratio test. 15. n=1 2n 4n - 3n n 2 . Convergent: use Limit Comparison with 4n Then use Ratio test, Root test or Geometric series. 2 ...
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This note was uploaded on 05/08/2008 for the course M 408 L taught by Professor Cepparo during the Spring '08 term at University of Texas at Austin.

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SampleSeries1Solns - SAMPLE PROBLEMS In each of the following problems decide if the series converges or diverges and specify which test(s you are

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