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Numerade Educator



Problem 22 Medium Difficulty

Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.

$ \displaystyle \sum_{n = 1}^{\infty} (-1)^{n-1} \frac {n}{8^n} $


$s \approx 0.0987, \quad n=5$


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Video Transcript

Let's graph the sequence of terms that's a M, and then the sequence of partial sums Guns Essen on the same screen. So here in does Mose and Purple, you could see I have my end. This is our formula for AM So the purple graph is the graph of the sequence of terms. And then down here in the red have the sequence of partial sons those air in red brass above. It looks like the summers more or less conversion to about point one, So that's fine. Rough estimate. So rough estimate from the graph. Now let's use the alternating Siri's estimation, Darryl. So notice that this Siri's here. This is alternating and B m, and over eight of them is positive. The limited bien a zero You could use Low Patel's rule for this one, and lastly, BN is decreasing, and the way to see that this is true is to define ffx to just be ex over eight to the ex and then go ahead and look at the derivative of F, and that numerator will be. So let's go. Maybe simplify this so the denominators positive, but the numerator will be negative if X is bigger than one over natural log of hey, more or less point five. So that shows that Sosa efforts decreasing this be sequence is decreasing, so we just showed decreasing. So this Siri's converges by the altering Siri's test. This is what allows us to use the alternating serious estimation zero and we'd like to be correcto four decimal places. So let's go to the next page. So now I see that b five more or less. This is larger than the desire twenty, however, be six six operates of the six, this one will be smaller, so more or less Let me go further here. Eight nine, This is less than so. This tells us that the reason will use five terms. So let me write the entire sum here. This's the infinite sum. Now, this will be approximately as five. Now, the reason I'm using five year instead of six is because the air is less than or equal to B in plus one less than zero point zero zero zero one. So the errors when using in terms. So here we had the smallest end that made this true. Was any pool six. I mean, excuse me, and plus one equals six. So that just means and equals five. So all that says there is that we just need five terms if we want to be correct with to the fourth decimal place. So here all we have to do is you go to a calculator and find this some right here. So let me go on to the next page right out of the sun. That's just one over eight of the one minus two over. It's where. Three over a cube for Andre to the fore, five over eight to the fifth and approximating that in a calculator. Zero point zero nine eight seven eight five four. But sense in this case, we're going to the fourth decimal place, so we should use point zero nine eight. So that'LL be our approximations of four decibels, and that's your final answer.