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In a Nut Shell:
An
alternating series is one in which every other term in the series
changes sign. The typical form is as follows:
∞ ∞
∑(-1)n+1
an or ∑(-1)n an
n = 1 n = 0
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There are several tests
that apply to alternating series.
The basic one is called the
alternating series test as follows:
If the alternating
series, ∑(-1)n+1
an satisfies the two
conditions:
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an ≥ an+1 > 0 for all n
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lim an = 0
n → ∞
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Then the alternating
series converges. Else, this test
does not apply.
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Theorem: If the
alternating series is absolutely convergent, then it is convergent.
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Three other tests apply to both alternating series and to
positive term series. As given
below.
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nth term test -
If lim
an ≠ 0 or does not exist, series diverges .
n →
∞
If lim an
= 0 , then there is no conclusion. The series may converge or diverge.
n → ∞
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Ratio Test
- If P= lim | (an+1)/ an
| exists or is infinite the series
n →
∞
∑ an converges absolutely if P < 1,
diverges if P > 1.
If P = 1 this test fails.
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Root Test –
Suppose P = lim |an| 1/n exists or
is infinite, then
n → ∞
the series ∑ an
a.
Converges absolutely if P
< 1
b.
Diverges if P > 1
c.
Test fails if P = 1
Click here to continue
with a discussion of conditional convergence.
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