General Contents
Detailed Contents
Index
Programmed tutorial: Absolute Convergence: Example 1
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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
Determine whether the series
converges or diverges.
General Contents
Detailed Contents
Index
Let’s try to visualize this series.
Are the terms all positive?
No, the cosine function varies between –1 and 1.
To check whether this is an alternating series, determine the signs of the first two terms.
Since the first two terms involve
cos
(
2) = -0.42 and
cos
(4) = -0.65,
we see that the series is not alternating.
In order to visualize this series, plot its terms on paper.
Then check your plot by clicking “Next”.
alt="Java applet graph of the terms of the absolutely convergent series." Your browser is completely ignoring the <APPLET> tag!
To get further
insight,
plot the sequence of partial sums on paper.
Then check by clicking “Next”.
alt="Java applet graph of the partial sums of the absolutely convergent series." Your browser is completely ignoring the <APPLET> tag!
Does the series appear to converge?
Yes, it appears to converge to –0.45.
What method can we use here?
We can test for Absolute Convergence.
Describe this test in terms of
a
n
.
If
converges,
also converges.
What will our
look like?
How does
compare with
?
Since the numerator of
is always ≤ 1,
we see that
What type of series
is
?
It is a p-series.
What is the value of p?
p is equal to 3 here.
Is this series convergent or divergent?
Since p > 1, this p-series is convergent.
Summarize our analysis so far.
We have shown that the terms of
are smaller than or equal to the terms of the
convergent p-series
.
What can we conclude about the
series
?
By the Comparison Test, this series converges.
What can we conclude about the original series,
, which doesn’t have the absolute value signs?
By the absolute Convergence Theorem, the original series,
, also converges.
Does
this agree
with the diagram above?
Yes.
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