Calculus II · Unit 4A · lesson
Alternating Series
Use decreasing magnitudes and a zero term limit to prove convergence through shrinking odd-even partial-sum brackets.
Section overview
Alternating, absolute, ratio, and root testsWhat this section is building
Use decreasing magnitudes and a zero term limit to prove convergence through shrinking odd-even partial-sum brackets.
Absolute convergence controls magnitude strongly enough to imply convergence; conditional convergence relies on cancellation.
Test absolute values first when practical, use the alternating-series hypotheses explicitly, and reserve ratio or root tests for matching algebraic structure.
Calling any alternating-looking series convergent or treating a ratio/root limit of one as a verdict.
Learning objectives
apply the Alternating Series Test and distinguish sign alternation from decreasing magnitude.
Alternating Series
Cancellation can rescue a divergent positive series
A positive series may diverge because every term pushes partial sums in the same direction. Alternating signs can create cancellation: one partial sum overshoots, the next undershoots, and the oscillation narrows as term magnitudes shrink. This mechanism can produce convergence even when the corresponding positive series diverges.
Alternation alone is not enough. The magnitudes must eventually decrease and approach zero. These hypotheses make odd and even partial sums monotone in opposite directions and force them toward a common limit. The test establishes convergence but usually not the exact sum.
Three separate alternating-series questions
First verify that the magnitudes eventually decrease. Second verify that those magnitudes approach zero. That proves convergence by the Alternating Series Test. Only after that should you test to decide whether the convergence is absolute or conditional.
Alternation works only when the overshoots shrink
An alternating series can converge because its partial sums approach the target from opposite sides. Odd partial sums form one monotone subsequence, even partial sums form another, and decreasing term magnitudes squeeze the two subsequences together.
Both conditions matter: the magnitudes must eventually decrease and must approach zero. Alternating signs alone create no guarantee. The theorem explains convergence through a geometric bracketing mechanism rather than through vague claims that positive and negative terms “cancel out.”
Read this graph as text
Alternating partial sums form shrinking brackets. Odd and even partial sums lie on opposite sides of the limit and the distance between them equals the next term magnitude. Odd and even partial sums nesting around a common limit. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in alternating partial sums form shrinking brackets; color is never the only cue.
Why it matters: Odd and even partial sums nesting around a common limit.
Odd and even partial sums lie on opposite sides of the limit and the distance between them equals the next term magnitude.
Alternating partial sums form shrinking brackets. Odd and even partial sums nesting around a common limit.
Two subsequences close in on the same number
With decreasing , even partial sums move in one direction and odd partial sums in the other. Their difference is the next term magnitude, which tends to zero, so both subsequences converge to the same limit.
Read this graph as text
Alternating partial sums bracket the limit. When alternating term magnitudes decrease to zero, odd and even partial sums approach the same value from opposite sides. Odd and even partial sums nesting around a common limit. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in alternating partial sums bracket the limit; color is never the only cue.
Why it matters: Odd and even partial sums nesting around a common limit.
When alternating term magnitudes decrease to zero, odd and even partial sums approach the same value from opposite sides.
Alternating partial sums bracket the limit. Odd and even partial sums nesting around a common limit.
Odd partial sums decrease while even partial sums increase. The next omitted term bounds the remaining vertical gap to the limit.
Alternating Series Test
If , eventually, and , then
converges.
The alternating harmonic series
For , the magnitudes decrease to zero. Therefore
converges, even though diverges.
Decrease may begin after a few terms
Test
Let . We have . For ,
so is decreasing for all sufficiently large . Therefore the series converges by the Alternating Series Test.
Check magnitudes, not signed terms
The sequence is usually not decreasing. The theorem requires the positive magnitudes to decrease eventually.
u4a-alternating_series_test-01Does converge?
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Show hint
Check decreasing magnitudes and the limit of .
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Classify .
Explain why diverges.
Find an alternating series for which magnitudes are not initially decreasing but eventually are.
Describe the behavior of odd and even partial sums.
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