InsallW98T3ps

InsallW98T3ps - 4. (20 pts.) Series II: For each of the...

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Insall - WS98 Calculus II Exam #3 March 24, 1998 1. (20 pts.) Theorems and Techniques: (a) Explain how to use the root test. (b) Complete the statement of the following theorem: If the sequence a does not converge to 0, then the series X n = n 0 a n ... 2. (20 pts.) Sequences: Answer the following. (a) The Fibonacci sequence is defined recursively by a 1 =1, a 2 = 1 and a n = a n - 1 + a n - 2 , for n 3. Find its first five terms. Does it converge or diverge? If it converges, what does it converge to? If it diverges, what is the nature of its divergence? (b) What is the limit of a if a n = n ! ( n +2)! ? 3. (20 pts.) Series I: (a) Explain when a geometric series converges or diverges. When it converges, explain how to compute its value. (b) Determine whether the series X n =1 ne - n 2 converges or diverges. If it converges, approximate its sum using the first 100 terms. Estimate the error in the approximation.
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Unformatted text preview: 4. (20 pts.) Series II: For each of the following, tell whether the given series converges absolutely, converges conditionally, or diverges. If it converges, approximate its value, explain how you approximated its value and give an estimate of the error in your approximation. (Remember to explain how you know whether the series converges or diverges.) (a) X n =1 5 2 n 3 + 1 (b) X n =5 (-1) n 1 n + 3 5. (20 pts.) Proofs: Prove the following: (a) X n =1 1 (3 n-2)(3 n + 1) = 1 3 (b) The series X n =1 (-1) n +1 2 n + 1 converges conditionally. The number of terms required to approximate its value to within 10-12 is at least 5(10 11 )....
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