Problem: Approximate the sum of the series

by its partial sum s₂₀ and estimate the error of this approximation.

Solution: It is easy to show using for instance the Ratio test or the Root test that the given series converges (an almost identical example can be found in the section Root and ratio test in Theory - Testing convergence), so the question makes sense.

My calculator says that

The error of approximation is

The most straightforward way to estimate the remainder in a series is to apply the Integral test. Using derivative we easily prove that the function f (x) = x⋅5^−x is decreasing on the interval [21,∞), it is also positive, so the Integral test applies. Using integration by parts and l'Hôpital's rule we get

Thus we have

Alternative: The above estimate using integral is very sensitive, just a little change in the series produces a function that we cannot hope to integrate. In general we therefore often do something different: We look for an upper estimate for the given series by another series that would be similar, convergent and that we could somehow sum up or easily estimate (for instance using integral). What estimates could we use for our given series?

In a typical series there are often parts in its terms that can be ignored for large values of k (see intuitive reasoning in Sequences - Theory - Limit), this then guides us in looking for a good upper estimate. However, this is not the case here, so we have to be more creative. One series that we know well is the geometric series. In order to get an upper estimate of the given series by a geometric series, we would need such an estimate for k. But since terms q^k grow faster than k if q > 1, this should not be a problem. We will obviously try to make q as small as possible, and we definitely need it to be smaller than 2, because the resulting series will have q/2 in it.

For instance, for k > 20 we have k < (1.5)^k. How do we know this for sure? We use the method outlined in Using derivative for comparing functions in Derivatives - Methods Survey. Denote f (x) = x and g(x) = (1.5)^x. Then f (20) < g(20) and f ′(x) < g′(x) for all x > 20, which proves our claim.

Thus we are justified to estimate

This is somewhat worse than the previous estimate, but it is not that bad and unlike the method above which is so sensitive to integration, this approach has a really good chance of succeeding when somebody throws a series at us. We could improve the quality of this estimate by tightening the upper bound, that is, using a number smaller than 1.5. A little experimentation shows that also k < (1.2)^k for k > 20, thus we get

This is almost as good as our first estimate. Check that when you use 1.17 instead of 1.2, you get the error estimate down to 7.40⋅10⁻¹⁴, which is almost perfect. If you want, you can play some more, but it since our first estimate is really tight, you can only hope to move from 7.4 to 7, the order is already the best one can get. By the way, how small a number can we use? The proof of the inequality with derivative that we used above shows that a^k is an upper estimate for k (for k > 20) if 20 ≤ a²⁰ and 1 ≤ ln(a)⋅a²⁰. If we use the first condition, we can replace the second one with an easier condition, ln(a) ≥ 1/20.

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