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 May 7th, 2016, 08:02 PM #1 Member   Joined: May 2015 From: Earth Posts: 64 Thanks: 0 Finding an approximation using series So the problem I am struggling with is attached, and is 8b. I am to use a given power series to find the approximation of a an integral of said power series so that the error between the actual and approximated answer is less than .01 - I got through most of the problem, to the point where I am left with the power series of -1/(1/3)^n, but i'm not sure what to do next. The answer key (also in the same attached file) tells me that since (1/3)^5 is less than .01 my answer is s sub 4, but I don't understand that. First, why am I setting (1/3)^n to be less than .01, since my series involves a negative number. And secondly, since it was proven that when n is 5 the error is less than .01, why is my answer s sub 4? Thanks. May 7th, 2016, 09:14 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2654 Math Focus: Mainly analysis and algebra I assume that you mean that the series is $\sum \left(-\frac13\right)^n$ since what you wrote doesn't converge. Because the series is an alternating series, the series of partial sums $s_n = \sum \limits_{k=1}^n a_k$ has the property $s_1 \gt s_3 \gt \ldots S \ldots \gt s_4 \gt s_2$ (or the reverse - depending on the sign of $s_1$) where $S$ is the limit of the partial sums. The important thing to notice then is that for any $n$ we have $$s_n \gt S \gt s_{n+1} \quad \text{or} \quad s_{n+1} \gt S \gt s_{n}$$ Thus, you need only find the first $n$ such that $|s_{n}-s_{n+1}| \lt 0.01$. Now, if $|a_{k+1}| \lt 0.01$ then $|s_{n} - s_{n+1}| = |a_{k+1}| \lt 0.01$, so the job is done. Last edited by v8archie; May 7th, 2016 at 09:18 PM. Tags approximation, finding, series Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post sepoto Calculus 3 December 5th, 2013 04:26 AM aaron-math Calculus 1 November 28th, 2011 03:04 AM Paul4763 Calculus 7 April 29th, 2011 09:54 AM naspek Calculus 5 December 10th, 2009 03:27 PM swm Algebra 5 March 30th, 2009 04:26 PM

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