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1. Compare to a harmonic series
The applet shows the series
A useful way to pick a comparison series when the target series uses a rational expression is to divide the highest power of n in the numerator by the highest power of n in the denominator, which in this case yields
The table shows the ratio an/bn, which does seem to converge to 1. We can verify this:
The limit comparison test says that in this case, both converge or both diverge. Since we know that the harmonic series diverges, A must also diverge.
2. Compare to a geometric series
Select the second example from the drop down menu, showing
Use the same guidelines as before, but include the exponential term also:
The limit of the ratio seems to converge to 1 (the "undefined" in the table is due to the b terms getting so small that the algorithm thinks it is dividing by 0), which we can verify:
The limit comparison test says that in this case, both converge or both diverge. Since
the B series is a geometric series with r = 1/2, which we know converges, so A also converges.
Other 'Sequences and Series' topics
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and
where an >0 and bn > 0.
(i.e., if the ratio of the terms tends to a finite number as n goes to infinity), then both series converge or both series diverge. By picking a suitable B, usually a p-series, we can use this test to determine whether or not A converges.