Math · Calculus II · Worked example
Test an alternating series
Does 1 − 1/2 + 1/3 − 1/4 + ⋯ converge? How close is the sum of its first ten terms?
Check the conditions
The signs alternate, the sizes 1/n decrease, and 1/n approaches 0, so the alternating series test applies: the series converges.
Absolute or conditional
Without the signs it is the harmonic series, which diverges, so the convergence is conditional.
Watch the partial sums
They step above and below the sum in turn: 1, then 1/2, then 5/6, then 7/12.
Estimate the error
The first ten terms add to 0.6456, and the error is at most the first term left out, 1/11 ≈ 0.091. The sum is in fact ln 2 ≈ 0.6931, which is 0.0475 away.
Result
It converges conditionally; the first ten terms, 0.6456, are within 1/11 of the sum, ln 2 ≈ 0.6931.
Your turn
Does the series of (−1)ⁿ⁺¹/n² converge absolutely?
Show the answer and explanation
Yes.
Without the signs it is the series of 1/n², a p-series with p = 2 > 1, which converges.
Keep exploring
In Sequences & infinite series, the terms (−1)ⁿ⁺¹/n show the partial sums zigzagging: 1, 0.5, 0.833, 0.583, … narrowing toward 0.693.
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