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### Section 10.13 : Estimating the Value of a Series

3. Use the Alternating Series Test and $$n = 16$$ to estimate the value of $$\displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^n}n}}{{{n^2} + 1}}}$$.

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Start Solution

Since we are being asked to use the Alternating Series Test to estimate the value of the series we should first make sure that the Alternating Series Test can actually be used on this series.

First, note that the $${b_n}$$ for this series are,

$b_n = \frac{n}{{{n^2} + 1}}$

and they are positive and with a quick derivative we can see they are decreasing and so the Alternating Series Test can be used here.

Note that it is really important to test these conditions before proceeding with the problem. It doesn’t make any sense to use a test to estimate the value of a series if the test can’t be used on the series. We shouldn’t just assume that because we are being asked to use a test here that the test can actually be used!

Show Step 2

Let’s start off with the partial sum using $$n = 16$$. This is,

${s_{16}} = \sum\limits_{n = 2}^{16} {\frac{{{{\left( { - 1} \right)}^n}n}}{{{n^2} + 1}}} = 0.260554530$ Show Step 3

Now, we know, from the discussion in the notes, that an upper bound on the absolute value of the remainder (i.e. the error between the approximation and exact value) is nothing more than,

${b_{17}} = \frac{{17}}{{290}} = 0.058620690$ Show Step 4

So, we can estimate that the value of the series is,

$\require{bbox} \bbox[2pt,border:1px solid black]{{s \approx 0.260554530}}$

and the error on this estimate will be no more than 0.058620690.