We end here our study of tests to determine convergence. The back cover of this text contains a table summarizing the tests that one may find useful.
While series are worthy of study in and of themselves, our ultimate goal within calculus is the study of Power Series, which we will consider in the next section. We will use power series to create functions where the output is the result of an infinite summation. Gregory Hartman Virginia Military Institute. Therefore we can apply the Alternating Series Test and conclude this series converges. We do not immediately conclude that we cannot apply the Alternating Series Test.
The important lesson here is that as before, if a series fails to meet the criteria of the Alternating Series Test on only a finite number of terms, we can still apply the test. Keep in mind that this does not mean we conclude the series diverges; in fact, it does converge. We are just unable to conclude this based on Theorem We can use Part 2 of the theorem to obtain an even more accurate result.
This cannot be solved algebraically, so we will use Newton's Method to approximate a solution. Convergence improvement can be effected by forming a linear combination with a series whose sum is known. Useful sums include. A general technique that can be used to acceleration converge of series is to expand them in a Taylor series about infinity and interchange the order of summation.
In cases where a symbolic form for the Taylor series can be found, this come sometimes even allow the sum over the original variable to be done symbolically.
For example, consider the case of the sum. The summand can be expanded about infinity to get. In particular, can be written. The application of this transformation can be efficiently carried out using Wynn's epsilon method. Letting , , and. The values of are there equivalent to the results of applying transformations to the sequence Hamming , p.
The transformed series exhibits geometric convergence. And once again, I'm not vigorously proving it here. Or I should say I'm not rigorously proving it over here. But the giveaway is that we have the same degree in the numerator and the denominator.
So now let's look at this one right over here. So here in the numerator I have e to the n power. And here I have e times n. So this grows much faster. I mean, this is e to the n power. Imagine if when you have this as , e to the th power is a ginormous number. Grows much faster than this right over here.
So this thing is just going to balloon. This is going to go to infinity. So we could say this diverges. Now let's look at this one right over here. Well, we have a higher degree term. We have a higher degree in the numerator than we have in the denominator.
So for the same reason as the b sub n sequence, this thing is going to diverge. The numerator is going to grow much faster than the denominator. In general finding a formula for the general term in the sequence of partial sums is a very difficult process. We will continue with a few more examples however, since this is technically how we determine convergence and the value of a series. This is actually one of the few series in which we are able to determine a formula for the general term in the sequence of partial fractions.
Therefore, the series also diverges. Again, do not worry about knowing this formula. The sequence of partial sums is convergent and so the series will also be convergent. The value of the series is,. As we already noted, do not get excited about determining the general formula for the sequence of partial sums. Two of the series converged and two diverged. Notice that for the two series that converged the series term itself was zero in the limit. This will always be true for convergent series and leads to the following theorem.
Then the partial sums are,. Be careful to not misuse this theorem! This theorem gives us a requirement for convergence but not a guarantee of convergence. In other words, the converse is NOT true.
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