Hi all,

Last week, we wrapped up integration with improper integrals and reviewed for the test this week on Tuesday and Wednesday. After the test, we will start with infinite sequences.

Improper integrals are all definite integrals that have singularities: one of the integration limits being infinity, the function is not defined at some point in the interval of integration or both. Improper integrals are defined as the limit of a normal definite integral, where the limit is taken for the integration interval to approach to the original interval with singularities. By definition, to evaluate an improper integral, you should follow two steps: evaluate a normal definite integral with parameters, then take the limit.

Sometimes all we care about is whether the improper integral converges and not the specific value of it, we can use the comparison rule. Given two non-negative functions f(x) and g(x), if we know f(x)>=g(x), then the improper integral of f converges implies the improper integral of g converges; the improper integral of g diverges implies that the improper integral of f diverges. A lot of the time, we use x^{-\alpha} as the comparison function, and \int_1^{\infity} x^{-\alpha}dx converges if and only if \alpha>=1. Similarly, \int_0^{1} x^{-\alpha}dx converges if and only if \alpha<1.

We will talk about infinite sequence on Thursday/Friday, and please review limits from Calc 1. We have already seen sequences when we studied Riemann sums, but we will study more general sequences in Chapter 11 and their properties.

Alright, that’s it for now. Sorry the sum up came later this week!

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