Hippocampus Calculus: Estimating the sum of a series

Suppose the series is convergent. You can use any partial sum as an approximation for S, the sum of this series.

To determine how good an approximation the nth partial sum is, we need to estimate the amount of error. This error, which is the difference between S and Sn, is called the remainder, Rn

Using reasoning similar to that used in the integral test, you know that the areas of the rectangles shown represent Rn. Comparing the area under the curve and the areas of the rectangles, you can see that and .

So, we can use the integral test to estimate the error in the remainder.

Consider the series . You can use your calculator to find the sum of the first five terms of this series. This can be considered an estimate of the sum of the series. But how accurate is this estimate? According to the remainder estimate theorem, . So, the size of the error is at most .00674.

How many terms are required to ensure that the sum is accurate to within 0.00001? To answer this question, we need to find the value of n for which Rn Â£ 0.00001.

, and it can be shown that . If e-n Â£ 0.00001, then Rn must be less than or equal to 0.00001 as well. Solving this inequality, we find that n Â³ 11.5. So we need 12 terms to ensure accuracy to within 0.00001.

We know that Rn is between and . If we add Sn to both sides of this inequality, we see that the sum of the series is between and because Rn + Sn = S. This inequality gives a lower and upper bound for S.

Let's see how we can estimate the sum of the series using n = 5. We saw that . We also know that . So, the sum is between S5 + e-6 and S5 + e-5. Therefore S is between 0.580579 and 0.584838.

If we consider the midpoint of this interval to be an approximation for S, the error is at most half the length of the interval. So, with error < 0.002.

Use the sum of the first 50 terms to estimate . Estimate the error involved in this approximation. Click "Submit" when finished.