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.
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.