Alternating Series

The Alternating Series Test

Suppose that a weight from a spring is released.  Let a1 be the distance that the spring drops on the first bounce.  Let a2 be the amount the weight travels up the first time.  Let a3  be the amount the weight travels on the way down for the second trip. Let a4 be the amount that the weight travels on the way up for the second trip, etc. Then  eventually the weight will come to rest somewhere in the middle. This leads us to

 Theorem: The Alternating Series Test    Let an  >  0 for all n and suppose that the following two conditions hold: { an } is a decreasing sequence for large n. Then the corresponding series and converge.

Proof:

We will prove the theorem for the second given series.  This is enough, since the first can be obtained from the second just by multiplying by -1.  We look at the series as adding two at a time and then adding them all together.

s2n = (a1 - a2) + (a3 - a4) + ...+ (a2n-1 - a2n)  >  0

which shows that this is bounded below by 0.  Now single out the first term and then add the rest two at a time

s2n = a1 - (a2 - a3) - (a4 - a5) - ...- (a2n-2 - a2n-1) - a2n

= a1 - [(a2 - a3) + (a4 - a5) + ...+ (a2n-2 - a2n-1) + a2n]

This second equation subtracts a positive number from the first term.  Hence

s2n < a1

which shows that the sequence is bounded above by a1.  Notice that s2n is monotonic since each difference is positive.  Therefore s2n  is bounded and monotonic and thus converges.  Since the an tend toward zero as n tends towards infinity, we have The limit of the partial sums exist and hence the series converges.

Example converges by the alternating series test, since the and

1                1
>
n              n + 1

Exercises:

Determine whether the following converge:

1. 2. The Remainder Theorem

Consider the spring example again.  The weight will always be between the two previous positions.  Hence we have

 The Remainder Theorem Let then           |L - sn|  <  an + 1

This says that the error in using n terms to approximate an alternating series is always less then the n + 1st term.

Example

Use a calculator to determine With an error of less than .01.

Solution:

We have

Error < .01

so choose n such that

1
<  .01
n

Here, n  =  101 will work.  Then use your calculator to get  0.70.

Absolute and Conditional Convergence

 Definitions is called absolutely convergent if converges. is called conditionally convergent if converges, but diverges.

Example:

The alternating harmonic series  is conditionally convergent since we saw before that it converges by the alternating series test but its absolute value (the harmonic series) diverges.

The series is absolutely convergent since the series of the absolute value of its terms is a P-series with p  =  2, hence converges.

The Rearrangement Theorem

 The Rearrangement Theorem Let be a conditionally convergent series and let k be a real number.  Then there exists a rearrangement of the terms so that you add them up and end up with k. As strange as it may seem, addition is not commutative for conditionally convergent series.  On the other hand for absolutely convergent series any rearrangement produces the same limit.