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With a spreadsheet or program it is quite easy to compute values
of the partial sums of a series. When a series has a ratio of successive terms
that is less than some r that is less than 1, the terms decrease exponentially
in r as j increases, and there is rarely a problem in computing value of the series
to any desired accuracy.
When the ratio of successive terms becomes close to 1 convergence becomes slower,
and one can hope to improve the accuracy of computations by fiddling with them.
When the rate of convergence of the jth partial sum behaves as j-k
for some power k, you can improve convergence by extrapolating. One way to do
this, as already noted in a number of contexts, is to replace the partial sum
which can also be written as 
When the coefficients of the series are standard functions of j this often happens
for integer values of k. You can easily examine ratios of the differences of terms
whose indices differ by a factor of 2 in order to see how the partial sums are
converging, and to choose a suitable k to extrapolate with. Doing this successively
allows computations to increase very greatly in accuracy.
You can even determine the rate of growth of the partial sums of a divergent
series by similar extrapolation.
For example, consider the sum of j2, which obviously diverges.
You can make the following deductions by looking at the partial sums sj
for j of the form 2k.
1. the terms increase by a factor of roughly 8 as j doubles. This indicates that
the leading term is proportional to n3 (which you probably should have
known)
2. by looking at the ratio of partial sums to n3 and extrapolating
you can see that the coefficient of n3 is 1/3.
3. by taking the differences between the partial sums 
you can produce another sequence whose ratios increase by roughly 4 as n doubles.
4. looking at the ratios of these to n2 you can find the coefficient
of n2 which is 1/2.
5. by looking at the differences of partial sums with 
you can find that it is 
Thus you can find the formula:
for this series, up to the nth term by numerical spreadsheet manipulations
alone.
There are, no doubt easier ways to get this result, but this one is really quite
painless if you are used to spreadsheet manipulations.
When the series does converge you can extrapolate to get accurate answers with
very little effort this way. By looking at powers of 2 you can first get rid of
the difference between the partial sum and the error that decreases by the leading
power of 2, then the next and the next etc... until the answer is accurate to
within your standard of same.
Exercises:
30.12 Do the steps above for the series n2 explicitly.
30.13 Find the sum of n-3 to ten place accuracy by appropriate
extrapolation.
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