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Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.$$\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}-\cdots$$

Calculus 2 / BC

Chapter 30

Expansion of Functions in Series

Section 1

Infinite Series

Series

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we want to find the first five partial sums and evaluate the convergence of the series given by one third minus 19 plus 1 27 21 plus one, 243rd and so on. First let's find the five partial some. So the rest of the partial some S. M. Is the sum from N equals one. The end of the A. M. Or the terms of our series, that's S. M. S minus one plus AM. So S one is therefore one third. The first term S two is S one minus +19 is two nights. S three is S two plus 1 27. 7 of 27. S four S three minus 1/81. 20/81. S five. S four plus one over 243 61 2043. Now to evaluate the conversions, let's think about whether or not our terms are absent owing to a particular value. 1/3 is 1/3. Two nights is a little bit smaller. 7/27 is getting really close to 1/4. 20 or 81 is getting even closer to 1/4. So if we use a calculator, we can confirm that the series convergence 2.25 or 1/4.

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