Download Lecture 21: Ratio and Root Tests (10.5) Recall Def. If ∑ |an

Lecture 21: Ratio and Root Tests (10.5)
Recall
X
Def. If
|an| converges, then we call
X
an absolutely convergent.
X
X
|an| diverges, but
an converges, then
X
we call
an conditionaly convergent.
If
We have many examples of absolutely convergent series, every convergent series in the Integral test and the comparison test lectures is an
absolutely convergent series.
We saw an alternating series can convergent, when
the non-alternating version does not. This kind
of series converge conditionally.
However, if a series is absolutely convergent, then
it converges, this is a stronger convergence.
1
X
Theorem: If a series
an is absolutely
convergent, then it is convergent.
ex. Does
∞
X
cos n
n=1
n2
converge absolutely?
Explain.
Ratio TestX
Let
an be a series and suppose that
an+1 , then
L = lim n→∞ an 1. If L < 1, the series converges absolutely,
2. If L > 1, the series diverges,
3. If L = 1, the test is inconclusive.
2
Notice that the Ratio test gives us more than
just convergence. If the ratio test shows a series
converges, then it converges absolutely.
The ratio test is most effective on series that contain a lot of multiplication. And if there is an
exponential or factorial term in your series, the
ratio test is a good first try to determine the
convergence.
(However, Ratio test does NOT work some series
including ’p−series’ like series–see NYTI example 3.).
Determine if the series converges absolutely or
conditionally:
1.
∞
X
3n
n=1
n!
3
2.
∞
X
2n
n=1
n2
1·3 1·3·5
n−1 1 · 3 · 5 · · · (2n − 1)
3. 1−
+
+· · ·+(−1)
+
3!
5!
(2n − 1)!
···
4
NYTI: Do the series converge absolutely?
Explain.
1.
∞
X
(−1)
2 n
n+1 n 2
n=1
2.
n!
∞
X
(−1)nn
n=1
2n
3. Show that both convergence and divergence
∞
X
are possible when L = 1 by considering
n2
and
∞
X
n=1
n−2 and
∞
X
n=3
n=1
ln n
.
n
When you work on homework problems, pay attention to which ones can’t be determined by
applying ratio test.
What do you do to determine the convergence of
these problems then?
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