**Exercise 1.** Decide on the convergence or divergence of

**Answer.** First notice that the denominator is equal to 0 when
*x*=1. Then the function inside the integral sign is unbounded at
*x*=1. Hence we have two bad points 1 and . So we must split
the integral and write

Let us first take care of the integral

We have

Since (by Bertrand's test) the improper integral

is convergent, then by the comparison test the improper integral

Next we take care of the integral

First notice that when , then

One may check this by showing that

On the other hand (and this is the crucial step in this exercise) is to find a polynomial approximation of when . This will be done via Taylor polynomials. Indeed, we have

when , which gives

Hence we have

which implies

when . The p-test implies that the improper integral

is convergent. Therefore the limit test implies that the improper integral

is convergent. Putting the two integrals together, we conclude that the improper integral

is convergent.

**Exercise 2.** Decide on the convergence or divergence of

**Answer.** The term is never equal for .
So let us focus on the term . According to the domain of the
tangent function, the only bad points we have to worry about is
. Clearly we have

Hence we have an improper integral or Type I at the bad point . Clearly we have

when . So let us approximate when . Again we will use Taylor polynomials, we have

when . This gives

when . Putting the stuff together we get

The p-test implies that the improper integral

is divergent. Therefore the improper integral

is divergent.

**Exercise 3.** Decide on the convergence or divergence of

**Answer.** The only improper behavior is around . Hence
this integral is of Type II not of Type I. Therefore no need for
splitting it. Note that when , then . Hence

when . The p-test implies that the improper integral

is convergent. Therefore the limit test implies that the improper integral

is convergent.

**Problem 1.** First decide on the convergence and divergence of

Then evaluate *f*(*x*).

**Problem 2.** Assume that is
continuous. Find , where

**Problem 3.** Consider the function

Find *f*(*x*).

**Problem 4.** Evaluate

**Problem 5.** In this problem, we will evaluate

**1**- Evaluate
**2**- Let which
function (that is
*f*is differentiable and*f*' is continuous). Show that **3**- Evaluate

**Problem 6.** Decide on convergence or divergence of

where *x* > 0 and is any real number.

**Problem 7.** Decide on convergence or divergence of

**Problem 8.** Decide on convergence or divergence of

If *I* is convergent, evaluate it.

**Problem 9.** Find

**
**

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Tue Dec 3 17:39:00 MST 1996

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