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# Prove an integral formula for ∑ (sin (nx)) / n2

Prove that the series

converges for all and let denote the value of this sum for each . Prove that is continuous for and prove that

Proof. First, the series converges for all real by the comparison test since

for all . Therefore, the convergence of implies the convergence of . Furthermore, this convergence is uniform by the Weierstrass -test with given by , and again converges. Thus, by Theorem 11.2 (page 425 of Apostol),

is continuous on the interval . Therefore, we may apply Theorem 11.4 (page 426 of Apostol):

since if and equals 2 if

# Show that we cannot interchange a limit and integral for fn(x) = nxe-nx2

For each positive integer define

Prove that the following limit and integral cannot be interchange:

Proof. First, we have

On the other hand,

Hence,

# Prove an inequality relating sums and integrals

Prove that

where is a nonnegative, increasing function defined for all .

Use this to deduce the inequalities

by taking .

Proof. Since is increasing we know (where denotes the greatest integer less than or equal to ) for all . Define step functions and by

Then and are constant on the open subintervals of the partition

So,

Since is integrable we must have

for every pair of step functions . Hence,

Next, if we take (which is nonnegative and increasing on ) we have

From this we then get two inequalities

Therefore,

# Find constants so the limit of an integral has a prescribed value

Find values for the constants and so that

First, since the integral is continuous (Theorem 3.4 of Apostol) we know

We also have

Hence, we may apply L’Hopital’s rule to the quotient giving us (using the First Fundamental Theorem of Calculus for the numerator)

Since the limit as of the numerator is 0, we must have the limit of the denominator equal to 0 as well (otherwise the whole limit would be 0 instead of 1).

Now, substituting back into our limit and using L’Hopital’s two more times (since again we have the indeterminate form ),

Therefore,

Hence, the requested constants are

# Prove or disprove given statements for functions such that f(x) = o(g(x))

Let and be functions, both differentiable in a neighborhood of 0, with and such that

Prove or disprove each the following statements.

1. as .
2. as .

1. True.
Proof. Since as we know by the definition of that

Thus, for every there exists a such that

So, for we have

The final line follows since by hypothesis. Therefore,

Hence,

By definition, we then have

2. False.
Consider for and for . Then, for ,

For we have .

Next,

Since we have as . However, since

does not exist.

# Use Taylor polynomials to approximate the integral of sin x / x

Using the Taylor polynomial approximation of find an approximation for the integral

Give an estimate for the error of the approximation. [Define when .]

We know (from this exercise, Section 7.8, Exercise #1) that the Taylor polynomial approximation of is given by

This implies

where

# Prove an inequality for the integral of 1 / (1 + x4)

Prove that

(Note: I cannot get the bounds Apostol asks for. I prove a different set below. I cannot figure out if it is a mistake in the book or not.)

Proof. Using the algebraic identity, valid for ,

we obtain

Therefore, integrating term by term,

Furthermore, we have

Taking , we then have

From the inequality for this integral we then have

# Prove an expression for the integral from 0 to 1 of (1 + x30) / (1+x60)

Prove that there exists a number with such that

Proof. For we have

(Since for we know .) Therefore, integrating the terms in the inequality from 0 to 1,

# Find an inverse for a function defined by an integral

Define a function for by

1. Prove that is strictly increasing on the nonnegative real axis.
2. If denotes the inverse of prove that is proportional to and find the constant of proportionality.

1. Proof. To show is strictly increasing we take the derivative,

Since for all we have that is strictly increasing on the nonnegative real axis

2. Proof. If is the inverse of then we know (Theorem 6.7 on page 252 of Apostol)

But, we have defined to the function such that . Therefore,

Therefore, we have that

Taking another derivative of with respect to we then have

# Prove some properties of the integral logarithm, Li (x)

The integral logarithm is defined for by

Prove the following properties of .

1. .
2. where is a constant depending on . Find the value of for each .
3. Prove there exists a constant such that

and find the value of this constant.

4. Let . Find an expression for

in terms of .

5. Define a function for by

Prove that

1. Proof. We derive this by integrating by parts. Let

Then we have

2. Proof. The proof is by induction. Starting with part (a) we have

To evaluate the integral in this expression we integrate by parts with

This gives us

Therefore we have

where . This is the case . Now, assume the formula hold for some integer . Then we have

We then evaluate the integral in this expression using integration by parts, as before, let

Therefore, we have

Plugging this back into the expression we had from the induction hypothesis we obtain

Therefore, the formula holds for the case , and hence, for all integers , where

and make the substitution , . This gives us . Therefore,

where is a constant

4. (Note: In the comments, tom correctly suggests an easier way to do this is to use part (c) along with translation and expansion/contraction of the integral. The way I have here works also, but requires an inspired choice of substitution.) We start with the given integral,

and make the substitution

Therefore, using the given fact that , we have

5. From part (d) we know that

Then, for the term we consider the integral

where . Similar to part (d) we make the substitution,

This gives us

Therefore, we have

Taking the derivative we then have