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In Section 8.2 of the text the sum of an infinite series is defined as , where *S*_{n} is the partial sum of the first *n* terms of the series. However, because of the algebraic difficulty (often, impossibility) of expressing *S*_{n} as a function of *n*, it is usually not possible to find sums by directly using the definition.

So, if we can generally not work from the definition, what can be done? The convergence tests of Sections 8.3, 8.4, 8.5 and 8.6 provide us with some needed tools. These are tests that tell us if a series converges, but - in the case the series does converge - do not tell us the sum of the series. Then, how do we find the sum? The answer is that we usually cannot find the sum. However, we can approximate the sum, *S*, by the partial sum *S*_{n}, for an appropriate value of *n*. But, what value of *n* will guarantee that *S* is, for instance, approximated by *S*_{n} to three decimal places?

For most of the tests there is not a good answer to this question. In many cases a heuristic approach is used - look at *S*_{n} for successive values of *n* until it seems that the third decimal place in the *S*_{n} will not change further. Note that there is no guarantee that this procedure will actually give the desired accuracy. For instance, it would be easy to be fooled by a slowly converging series.

For the convergent series , we define the remainder after *n* terms as . It should be clear that *S* = *S*_{n} + *R*_{n} and that *R*_{n} is in fact the error when *S*_{n} is used to approximate *S*. As is usual in approximation arguments, we seek an upper bound on the absolute value of the error. The argument that is used to prove the integral test can be modified so as to establish the following bounds.

where *f*(*i*) = *a*_{i} when *i* is a positive integer. Thus, under the assumption that *a*_{k} > 0 for all *k*, we have

That is, we have a bound on the error arising by using *S*_{n} to approximate *S*.

Use the Maple command **?help** to read more about the **sum** command. Note especially the use of single quotes when using **sum**.

Sometimes it may be advantageous to use **Sum** and **value** instead of just **sum**. The use of **Sum** enables you to check if you have typed the series correctly. As an example, look at

> Sum('1/2^k','k'=1..100);

> value(");Also try these commands with "100" replaced by "infinity." When you use

> evalf(sum( ));Otherwise, in some cases, Maple will attempt to do exact arithmetic and the answer may be long enough to fill several screens.

- 1.
- Use Maple to apply a comparison test to the given series. What is the behavior of the series? Explain. Hint: use .

- 2.
- Use Maple to apply the ratio test to the given series. What is the behavior of the series? Explain.

- 3.
- Use Maple to apply the integral test to show that the given series converges.

- 4.
- For the series of Exercise 3, use Maple to implement the error bound theory discussed in the Background section. In particular, find the smallest
*n*that guarantees*S*_{n}approximates*S*with an error less than 0.05. - 5.
- Find
*S*_{n}for the value of*n*found in Exercise 4. In fact, Maple can sum the series we have been discussing. Have Maple do so. How does*R*_{n}compare with ?. Did the error bound theory work well for this series?

1/20/1998