Unit 1: Sequences & Series TestsStudent ID:

Day 18 – TheAlternating Series TestError Bound

Opener:Graph the first ten partial sums of each series. Record the graph below. Then, approximate the sum.

Graph the first ten partial sums of the following sequence. Record the graph below. Then, approximate the sum.
/ Graph the first ten partial sums of the following sequence. Record the graph below. Then, approximate the sum.

Compare and contrast series (a) and (b) above.

Lesson:We know that series (b) from the opener is the ______Series. The first series (a) above is almost identical to series (b), but because the signs alternate, the series is (creatively) known as the ______.

Our last test for convergence is very easy. It is known as the ______Test. There are two criteria to test for, and they are below.

An alternating series in the form of

Converges given:
“Eventually” the terms of the sequence are nonincreasing
AND

Note: the Alternating Series Test only tells if a sequence converges. It does not tell divergence of a series!

Cliff’s Notes: perform the Divergence Test, and make sure the sequence decreases!

Determine whether the following series converge or diverge.

Alternating Series Error Bound

Let’s take a closer look at the sum of the Alternating Harmonic Series (remember series (a) from the opener). Use the number line to graph the progression of the first tenpartial sums of the Alternating Harmonic Series, beginning with . Use arcs to follow the progression, and label them with where is the partial sum.

Realize that to find the actual sum of this series, we would have to add up an infinite number of terms. However, this is not realistic; we must stop at some partial sum known as . Of course, that means we’ve left off some terms at the end, which we call the remainder or . Therefore, the actual sum of any series is:

Of course, using Algebra, we can manipulate our above equation to get a relationship for the remainder:

With Alternating Series, though, we put the remainder equation in absolute value so that we can find the magnitude of the remainder, instead of having the worry about the sign. This gives us:

Look back to the number line. Notice that the next term in the sequence always pushes the current sum PAST the actual sum (either above or below, though potentially equal). This means, instead of dealing with the vague remainder ,we can relate the actual value of an alternating series to the sum after partial sums using the next “neglected” term, which is to say .

Therefore, we can get a better idea of the remainder for an alternating series. By the two relationships above:

In plain terms, the remainder of an alternating series is always less than, or equal to, the next neglected term in a partial sum. In fact, we say that the remainder of the series is ______by . Hence, the Alternating Series Error Bound. Let’s see how we can use this:

Find for the Alternating Harmonic Series given in part (a) of the opener / What is the maximum amount that the partial sum differs from the actual sum of the Alternating Harmonic Series?
What is the maximum amount that the partial sum differs from actual sum of the Alternating Harmonic Series? / How many terms are required to approximate the Alternating Harmonic Series with a maximum error less than ?

As a concluding note, it turns out that the Alternating Harmonic Series is exactly equal to . Prove it to yourself using a calculator. More on WHY later!

Bookwork: p.659 #s 2, 11 – 37 odds, 39 – 42 (calculator ok)