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Example Questions
Example Question #2951 : Calculus Ii
Determine the convergence of the series using the Comparison Test.
Series converges
Series diverges
Cannot be determined
Series diverges
We compare this series to the series
Because
for
it follows that
for
This implies
Because the series on the right has a degree of equal to
in the denominator,
the series on the right diverges
making
diverge as well.
Example Question #2952 : Calculus Ii
Does the series converge or diverge? If it does converge, then what value does it converge to?
Diverges
Converges to 1
Converges to
Converges to
Converges to
Converges to 1
To show this series converges, we use direct comparison with
,
which converges by the p-series test with .
Thus we must show that
.
Cross multiplying the previous section and multiplying by
, we obtain
.
Since this holds for all we can conclude that
.
Summing from to
, and noting that
for all
, we obtain the following inequality:
.
Therefore the series
converges by direct comparison.
Now to find the value, we note that
,
so that
.
Now let
be a sequence of partial sums.
Then we have
Therefore
.
Taking the limit as , we obtain the following:
Therefore we have
.
Example Question #2953 : Calculus Ii
Does the series converge?
Cannot be determined
Yes
No
No
Notice that for
This implies that
for
Which then implies
Since the right-hand side is the harmonic series, we have
and thus the series does NOT converge.
Example Question #2954 : Calculus Ii
Determine whether the series converges, absolutely, conditionally or in an interval.
Converges absolutely
Does not converge at all
Converges in an interval
Converges conditionally
Converges absolutely
Example Question #2955 : Calculus Ii
Determine whether the series converges
Converges in an interval
Does not converge at all
Converges absolutely
Converges conditionally
Converges absolutely
Example Question #2956 : Calculus Ii
Test for convergence
Converges conditionally
Cannot be determined
Converges in an interval
Converges absolutely
Diverges
Converges absolutely
Step 1: Recall the convergence rule of the power series:
According to the convergence rule of the power series….
converges as long as
Step 2: Compare the exponent:
Since , it is greater than
. Hence the series converges.
Step 3: Conclusion of the convergence rule
Now, notice that the series isn’t an alternating series, so it doesn’t matter whether we check for absolute or conditional convergence.
Example Question #2957 : Calculus Ii
Test for convergence
Can’t be determined
Diverges
Converges absolutely
Converges Conditionally
Converges in an interval
Converges absolutely
Step 1: Try and look for another function that is similar to the original function:
looks like
Step 2: We will now Use the Limit Comparison test
Since the limit calculated, is not equal to 0, the given series converges by limit comparison test
Example Question #2958 : Calculus Ii
Does the following series converge or diverge?
Absolutely converge
Conditionally converge
Diverge
The series either absolutely converges, conditionally converges, or diverges.
Absolutely converge
The best way to answer this question would be by comparing the series to another series,, that greatly resembles the behavior of the original series,
. The behavior is determined by the terms of the numerator and the denominator that approach infinity at the quickest rate. In this case:
When this series is simplifies, it simplifies to a series that converges because of the p-test where .
With two series and the confirmed convergence of one of those series, the limit comparison test can be applied to test for the convergence or divergence of the original series. The limit comparison test states that two series will converge or diverge together if:
Specifically:
This limit equals one because of the fact that:
if the coefficients come from the same power.
Because the limit is larger than zero, and
will converge or diverge together. Since it was already established that
converges, the original seies,
, converges by the limit comparison test.
Example Question #2959 : Calculus Ii
Determine if the following series converges or diverges:
Series diverges
Series converges
Series converges
for all
;
is a sum of geometric sequence with base 1/3.
Therefore, said sum converges.
Then, by comparison test, also converges.
Example Question #2960 : Calculus Ii
What can be said about the convergence of the series ?
Diverges
Inconclusive
Converges
Converges
Since for all n>0, and
converges as a P-Series, we may conclude that
must also converge by the comparison test.
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