Problem: Decide whether the following integral converges:
In this integral there are two problems, namely each endpoint. Infinity is
We will now look at convergence of the two integrals, starting with the latter. Since this looks like a very simple integral, we will try to evaluate it:
(We already know that the last integral is divergent). This means that there is no need to investigate the second part (the integral from 1 to 2), the whole given integral is divergent.
1. Note that the convergence of the integral we evaluated above cannot be decided by our tests. First, try the comparison. For
and the Comparison test is inconclusive.
We could try to compare the given function to some powers that have
convergent integrals to infinity, but this is not possible. This should
become clear when we attempt limit comparison. The test function
If we try a higher power in the denominator, the function will be too small.
Precisely, for any
Thus powers do not help as test functions in the Limit Comparison test.
Now it is useful to recall the scale of powers at infinity. While
converges exactly if
2. Although it was not necessary to check on the integral from 1 to 2 to solve our problem, it is nevertheless a good exercise. Since we can easily integrate, we directly evaluate:
So this integral is also divergent. However, logarithm behaves quite
differently around 1 as compared to infinity. For instance, while around
infinity, logarithm cannot be compared to any power (as we saw above), for
values of x close to 1 we have
Justification now succeeds:
Since the test integral
diverges, by the Limit Comparison test, also the integral of f from 1 to 2 diverges.
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