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Mathematical Expression Editor

Exercises relating to the direct and limit comparison tests for series.

Suppose you wished to use the Direct Comparison Test to establish convergence of
the series Which of the following options below would be valid series for comparison?
Select all that apply.

Expand for feedback on :

This comparison wouldn’t be valid because which is the wrong direction for the
inequality when showing convergence.

Expand for feedback on :

This comparison is valid because which is the proper direction for the comparison
inequality when showing convergence. Also key is that the -series for is convergent.

Expand for feedback on :

This comparison is valid because which is the proper direction for the comparison
inequality when showing convergence. Also key is that the -series for is convergent.

Expand for feedback on :

This comparison is valid because which is the proper direction for the comparison
inequality when showing convergence. Also key is that the -series for is convergent.

Expand for feedback on :

This comparison is not valid because the series (known as the harmonic
series) is not convergent, so no comparisons to it can establish convergence.

Suppose you wished to use the Direct Comparison Test to establish convergence of
the series Which of the following options below would be valid series for comparison?
Select all that apply.

Expand for feedback on :

This comparison is not valid because which is the wrong direction for the
comparison inequality when showing convergence.

Expand for feedback on :

This comparison is valid because (because and is always greater than
whenever .) which is the proper direction for the comparison inequality
when showing convergence. Also key is that the -series for is convergent.

Expand for feedback on :

This comparison is valid because (because and is always greater than
whenever .) which is the proper direction for the comparison inequality
when showing convergence. Also key is that the -series for is convergent.

Using only the comparison functions listed here: which of the following series can be
proved to either diverge or converge using the Direct Comparison Test? (Select all
that apply.)

For the series :

This series is convergent, but none of the allowed comparisons will work: Only and
are larger than , but neither of those series converges.

For the series :

Comparison to works fine in this case.

For the series :

so , so actually diverges by direct comparison.

For the series :

This series textitis convergent, but none of the convergent comparison options and
is greater than .

For the series :

The direct comparison works fine to show convergence of the series.

For the series :

This is a divergent series, but direct comparison with the allowed options doesn’t
work because is smaller than the two divergent options and .

Suppose you wished to use the Limit Comparison Test to establish convergence of
the series Which of the following options below would be valid series for comparison?
Select all that apply.

Expand for feedback on :

This comparison wouldn’t be valid because is too small: the limit of divided by is
infinite, which is an inconclusive case.

Expand for feedback on :

This comparison is valid because the limit divided by is , which means that they
both converge (because a -series for is convergent).

Expand for feedback on :

This comparison is valid because is a convergent -series which is much larger than ,
i.e.,

Expand for feedback on :

Also works for the same reason as : it’s a convergent -series and the original series
divided by the comparison series goes to zero.

Expand for feedback on :

This comparison is not valid because the series (known as the harmonic
series) is not convergent, so no comparisons to it can establish convergence.

Suppose you wished to use the Limit Comparison Test to establish convergence of
the series Which of the following options below would be valid series for comparison?
Select all that apply.

For the Limit Comparison Test, non-dominant terms don’t matter, so the results will
be the same here as they were for the series .

Using only the comparison functions listed here: , which of the following series can
be proved to either diverge or converge using the Limit Comparison Test? (Select all
that apply.)

For the series :

This series is convergent, but none of the allowed comparisons will work: Only and
are larger than , but neither of those series converges.

For the series :

Comparison to works fine in this case.

For the series :

so , so the limit so we get divergence by the divergence of the comparison series.

For the series :

Here the non-dominant term in the denominator is no longer relevant, so limit
comparison to works fine.

For the series :

The direct comparison works fine to show convergence of the series.

For the series :

Limit comparison with works great: divided by equals