General Contents
Detailed Contents
Index
Programmed tutorial: p-Series Test of Convergence or Divergence of a Series: Example 2
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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
First, plot the terms of this series on paper.
To check, click “Next”.
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To get further insight, plot the sequence of partial sums on paper.
Then check by clicking “Next”.
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Does it appear that the series is converging?
Yes.
What type of series might this be?
Let’s see if it is a
p
-series.
What is the form of the general term of a
p
-series?
Write the general term of the given series.
So, this is a
p
-series.
Does it converge?
Yes.
Why?
A
p
-series converges only if p > 1.
Does this agree with our diagrams above?
Yes.
Lets estimate the remainder after 5 terms.
Write the general equation for the bounds on the remainder.
The remainder after summing
N
terms is
R
N
= S –
S
n
.
For a p-series,
R
N
is bounded by
Substitute to set up bounds on
R
5.
R
5
= S – S
5
.
R
5
is bounded by
Simplify to get a decimal.
How do we use this result?
We add it to
S
5
.
Since the remainder could be as small as 0.0,
Approximate
S
5
by calculating and adding the first 5 terms.
Use this result to state bounds for
S
.
, or
Does
this agree
with our diagram above?
Yes, the sequence of partial sums appears to approach approximately 1.3 in the diagram.
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