An alternating series is an infinite sum whose terms switch signs, such as positive, negative, positive, negative. These series are important because the sign changes can make a series converge even when the related positive-term series does not. The Alternating Series Test gives a simple way to prove convergence without finding the exact sum.
It also helps estimate how close a partial sum is to the true infinite sum.
The main idea is that if the term sizes steadily shrink toward zero, the partial sums overshoot and undershoot the limit by smaller and smaller amounts. On a graph, the partial sums form a zigzag path that closes in on one limiting value. If the conditions are met, the error after stopping at n terms is no larger than the size of the next term.
This makes alternating series useful for approximations in calculus, such as logarithm, arctangent, and power series calculations.
Understanding Calculus: The Alternating Series Test
The cancellation works because each new term pulls a running total back toward a central value. Imagine first adding a positive amount, then removing a slightly smaller amount. The result stays above the eventual total.
The next positive amount is smaller again, so it cannot push the total as far away as before. This repeating correction creates two sequences of partial sums. One sequence approaches the limit from above.
The other approaches it from below. The gap between them becomes tiny when the omitted terms become tiny.
There are two checks to make before using the test. First, ignore the signs and study the positive sizes of the terms. They must settle into a nonincreasing pattern.
A few early increases do not matter, since an infinite series is controlled by its long-term behavior. Second, the sizes must approach zero. This second condition is essential.
If the terms keep a noticeable size, then every new addition changes the total by a noticeable amount. A total that continues making fixed-size jumps cannot settle at one number. Reaching zero alone is not enough for every kind of series, but with the alternating pattern and eventual decrease, it is enough here.
Students should distinguish convergence from absolute convergence. Remove the signs from the alternating harmonic series and the resulting positive series grows without bound. Keeping the signs produces a finite total.
This is called conditional convergence. It is a special situation with an important warning. Rearranging the terms of a conditionally convergent series can change its total, or even make the rearranged series fail to converge.
In ordinary finite addition, order never matters. For infinite sums with positive and negative terms, order can matter greatly. Keep the given order unless a problem gives a reason to change it.
The error estimate turns a convergence proof into a practical calculation tool. To approximate a sum, calculate successive partial sums until the next omitted term is smaller than the allowed error. For example, if an answer must be within one thousandth, stop once the next term has size below one thousandth.
The sign of that next term tells which side of the true value the current partial sum lies on. Alternating power series appear when calculators and computer programs approximate functions such as sine, cosine, natural logarithm, and inverse tangent.
In class, write out the term size separately from its sign, check eventual decrease carefully, then state the limit of the sizes. This prevents a common mistake of noticing alternating signs and assuming that they guarantee convergence.
Key Facts
- An alternating series often has the form sum from n = 1 to infinity of (-1)^(n+1) b_n or (-1)^n b_n, where b_n > 0.
- Alternating Series Test: if b_(n+1) <= b_n for all sufficiently large n and lim as n -> infinity of b_n = 0, then sum (-1)^(n+1) b_n converges.
- The terms must decrease in size toward zero, but they do not need to decrease from the very first term if they eventually decrease.
- Alternating Series Error Bound: if S is the true sum and S_n is the nth partial sum, then |S - S_n| <= b_(n+1).
- For the alternating harmonic series, sum from n = 1 to infinity of (-1)^(n+1)/n converges by the test because 1/n decreases and lim as n -> infinity of 1/n = 0.
- The Alternating Series Test proves convergence, but it does not usually give the exact value of the infinite sum.
Vocabulary
- Alternating series
- An infinite series whose terms change sign back and forth.
- Partial sum
- The sum of the first n terms of an infinite series, usually written S_n.
- Convergence
- A series converges when its sequence of partial sums approaches a finite number.
- Remainder
- The remainder is the difference between the true infinite sum and a chosen partial sum.
- Monotone decreasing
- A sequence is monotone decreasing when each term is less than or equal to the previous term.
Common Mistakes to Avoid
- Forgetting to check that b_n approaches zero. A decreasing sequence of term sizes is not enough, because the terms must shrink to zero for the test to apply.
- Testing the signed terms instead of the positive term sizes. In the Alternating Series Test, b_n represents positive magnitudes, so the decreasing condition applies to b_n, not to (-1)^n b_n.
- Assuming convergence is absolute just because the alternating series converges. The Alternating Series Test only proves convergence of the signed series, and the related positive series may still diverge.
- Using the wrong term in the error bound. The error after S_n is bounded by the next unused term b_(n+1), not by b_n unless the indexing has been carefully adjusted.
Practice Questions
- 1 Determine whether the series sum from n = 1 to infinity of (-1)^(n+1)/(3n + 2) converges by the Alternating Series Test. State the two conditions you checked.
- 2 For the alternating harmonic series sum from n = 1 to infinity of (-1)^(n+1)/n, how many terms are needed to guarantee an error less than 0.01 when using S_n?
- 3 A student says sum from n = 1 to infinity of (-1)^n n/(n + 1) should converge because the signs alternate. Explain why this reasoning is incorrect using the conditions of the Alternating Series Test.