121616949-math.265 - and twelve terms which give 0 737 and...

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10.5 Comparison Tests 251 and that we approximate L by a finite part of this sum, say L N X n =0 ( - 1) n a n . Because the terms are decreasing in size, we know that the true value of L must be between this approximation and the next one, that is, between N X n =0 ( - 1) n a n and N +1 X n =0 ( - 1) n a n . Depending on whether N is odd or even, the second will be larger or smaller than the first. EXAMPLE 10.30 Approximate the alternating harmonic series to one decimal place. We need to go roughly to the point at which the next term to be added or subtracted is 1 / 10. Adding up the first nine and the first ten terms we get approximately 0 . 746 and 0 . 646. These are 1 / 10 apart, but it is not clear how the correct value would be rounded. It turns out that we are able to settle the question by computing the sums of the first eleven
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Unformatted text preview: and twelve terms, which give 0 . 737 and 0 . 653, so correct to one place the value is 0 . 7. Exercises Determine whether the following series converge or diverge. 1. ∞ X n =1 (-1) n-1 2 n + 5 ⇒ 2. ∞ X n =4 (-1) n-1 √ n-3 ⇒ 3. ∞ X n =1 (-1) n-1 n 3 n-2 ⇒ 4. ∞ X n =1 (-1) n-1 ln n n ⇒ 5. Approximate ∞ X n =1 (-1) n-1 1 n 3 to two decimal places. ⇒ 6. Approximate ∞ X n =1 (-1) n-1 1 n 4 to two decimal places. ⇒ 10.5 Comparison Tests As we begin to compile a list of convergent and divergent series, new ones can sometimes be analyzed. EXAMPLE 10.31 Does ∞ X n =2 1 n 2 ln n converge?...
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