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Section 8.1: Sequences Practice HW from Stewart Textbook (not to hand in) p. 565 # 3-33 odd
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Sequences Sequences are collection of numbers or objects that is ordered by the positive integers. Notation: (known as the terms of the sequence).
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Example 1: Write the first five terms of the sequence. Solution:
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Example 2: Write the first five terms of the sequence. Solution:
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Describing the nth term of a sequence Involves writing a formula describing the pattern the sequence of numbers follows.
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Example 3: Find a formula for the general term of the sequence. Solution:
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Example 4: Find a formula for the general term of the sequence. Solution:
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Convergence of Sequences Sequences whose behavior approaches a limiting value are said to converge to this value
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Example 5: What value does the sequence appear to converge to? Solution:
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Definition A sequence has the limit L and we write which is read approaches L as n approaches if we can make the terms as close to L as we like by taking n sufficiently large. If exists, then the sequence is convergent. Otherwise, it is divergent.
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Useful Facts for Showing Convergence of Sequences 1. L Hopital’s Rule for indeterminate forms. 2. Absolute Value Theorem: If, then. 3. Using Maple to graph.
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Example 6: Determine whether the sequence converges or diverges. If the sequence converges, find the limit. Solution:
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Example 7: Determine whether the sequence converges or diverges. If the sequence converges, find the limit. Solution:
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Example 8: Determine whether the sequence converges or diverges. If the sequence converges, find the limit. Solution:
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Example 9: Determine whether the sequence converges or diverges. If the sequence converges, find the limit. Solution:
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Example 10: Determine whether the sequence converges or diverges. If the sequence converges, find the limit. Solution: (In typewritten notes)
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Example 11: Determine whether the sequence converges or diverges. If the sequence converges, find the limit. Solution:
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Factorial Recall that, read as n factorial, is defined to be Note:.
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Example 12: Compute. Solution:
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Example 13: Determine whether the sequence converges or diverges. If the sequence converges, find the limit. Solution: (In typewritten notes)
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