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May 7th, 2016, 09:02 PM   #1
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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.
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May 7th, 2016, 10:14 PM   #2
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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 10:18 PM.
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