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hw-11-05-alternating - to find out Question B(optional The...

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Chapter 11.5 homework: Alternating Series The usual even/odd policy applies, except as noted. #1 (required, even though it is odd) Drill: #2 to #6: do these informally and quickly Take more care on these: #8 #9 #13 #14 #23 Question A: (optional) The book says that 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... converges to ln(2)=0.693ish. Using Excel, Compute the first 50 partial sums, and their errors: abs(s_n - log(2)) Then plot those errors to see that they decrease toward zero. We might also ask: how quickly do they decrease? Like 1/x? like 1/x^2? like exp(-x)? Ask me later how to do log-scale plots
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Unformatted text preview: to find out. Question B: (optional) The book's theorem on alternating series says: IF b_n is decreasing AND lim b_n = 0 THEN the series converges. It does *not* say: If either of those conditions fails, then the series diverges. i) Can you find a b_n (all terms >= 0 ) where it's sometimes increasing and sometimes decreasing (all the way to infinity), and the series (-1)^n b_n converges? ii) Can you find a b_n that is purely decreasing, but whose limit is not 0, for which the series still converges? iii) Any other similar explorations?...
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