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Approximate the sum of the series correct to four decimal places.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^{n-1}}{n 4^n} $

0.2232

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Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

it's approximate this some of this alternating serious. It's a four decimal places. So here, if we want to be correct up to four decimal places, we need the air to be less than I'LL put the less than way out here zero point zero zero zero zero five and explain that. So if the ear is lost in this decimal over here, so notice that if you're less than this, then when you round off the this number in the fifth spot is not large enough, Teo increase the four decimal. That means that we're currently four decimal places. That's the reason for this number, however, sense for dealing with the alternating Siri's the alternating Siri's estimation. Dirham says that the air, when approximating the sum by using in terms and the partial song, is less than being plus one or being is just the absolute value of a M. So bn is just one over end times for the and power. So we want this to be less than zero point zero zero zero zero five. That turns out to be true. Using a calculator, you could see that you want and to be six or more so here we have n plus one bigger than or equal to six so and bigger than equal So five. That means that in our approximation that we should just be using five terms. So here they're approximation will just be sn or as five, actually, because here, that's the number five is approximately so this you will go to the calculator. I'LL write this out in the exact form. So this is just the sum from one to five negative one and minus one over and four to the end and then going toe a calculator and plugging this in way into two and then three two. So that's our approximation of the infinite sums of four decimals, and that's our final answer.