What is alternating series test?

The Alternating Series Test is a method used to prove that a series is convergent. Specifically, it applies to <a href="https://www.wikiwhat.page/kavramlar/alternating%20series">alternating series</a>, which are series whose terms alternate in sign.

To apply the test, consider an alternating series of the form:

∑ (-1)^n * b_n or ∑ (-1)^(n+1) * b_n

where b_n > 0 for all n.

The Alternating Series Test states that the series converges if the following two conditions are met:

1. Decreasing terms: The sequence {b_n} is monotonically decreasing, meaning that b_(n+1) ≤ b_n for all n greater than some index N.

2. Limit to zero: The limit of the sequence {b_n} as n approaches infinity is zero, i.e., lim (n→∞) b_n = 0.

If both conditions are satisfied, the alternating series is convergent. If the second condition (limit to zero) fails, the series diverges by the <a href="https://www.wikiwhat.page/kavramlar/divergence%20test">divergence test</a>. If the terms don't decrease but the limit is zero, the test is inconclusive.

Important Notes:

• The test only proves convergence; it doesn't tell you what the series converges to.
• If the series is found to be convergent by the alternating series test then we can find an <a href="https://www.wikiwhat.page/kavramlar/error%20bound">error bound</a> when approximating the series by its partial sums. The error bound is simply the absolute value of the first neglected term.