Alternating Series Test
It's the test for Alternating series
.
►Refer to Khan academy: Alternating series test ►Refer to xaktly: Alternating Series
Alternating Series
It means, Terms of the series "alternate" between positive and negative.
etc., The alternating harmonic series
:
The Alternating Series Test
The very good example of this test is the Alternating Harmonic Series
:
▲ It does CONVERGES. (But the Harmonic Series does NOT converge)
Strategy:
Take AWAY the
Alternating sign (-1)ⁿ
:Determine if the rest part is a decreasing series:
Take limit of the rest part:
If
Limit = 0
, then the series CONVERGES.If
Limit ≠ 0
, then the series DIVERGES.
Example
Notice this is an
alternating series
, so we're to apply thealternating series test
.Take away the
alternating term
, and left with(2/p)ⁿ
.So the series only converges if
(2/p)ⁿ
is decreasing and its limit is0
.And the only way to make it decreasing is to make sure
(2/p) < 1
.Based on that
p
value, the limit of(2/p)ⁿ
is surely a0
.Therefore,
p > 2
makes the series converges.
Example
Notice this is an
alternating series
, so we're to apply thealternating series test
.Take away the
alternating term
, and left with(2n)ᴾ
.So the series only converges if
(2n)ᴾ
is decreasing and its limit is0
.And the only way to make it decreasing is to make sure
p < 0
.Based on that
p
value, the limit of(2n)ᴾ
is surely a0
.Therefore,
p < 0
makes the series converges.
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