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
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 qk
grow faster than k if
For instance, for
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
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