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# 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,

# Prove some statements about integrals of bounded monotonic increasing functions

Consider a bounded, monotonic, real-valued function on the interval . The define sequences

1. Prove that

and that

2. Prove that the two sequences and converge to .
3. State and prove a generalization of the above to interval .

1. Proof.First, we define two step functions,

where denotes the greatest integer less than or equal to . Then we define a partition of ,

For any we have

So, and are constant on the open subintervals of the partition .
Since is monotonically increasing and for all (by the definition of and ) we have

2. Proof. From part (a) we have

since . Since does not depend on we have

Therefore,

3. Claim: If is a real-valued function that is monotonic increasing and bounded on the interval , then

for and defined as follows:

Proof. Let

be a partition of the interval . Then, define step functions and with and for . By these definitions we have for all (since is monotonic increasing). Since is bounded and monotonic increasing it is integrable, and

And, since , and we have

# Sketch inequalities in the complex plane

Sketch each of the following sets of complex numbers that satisfy the given inequalities:

1. .
2. .
3. .
4. .

1. Letting we have,

This is a disk of radius centered at . The sketch is as follows:

2. Letting we have,

This is the half-plane with negative real part. The sketch is as follows:

3. Letting we have,

This is the half-plane with positive imaginary part. The sketch is as follows:

4. Letting we have,

This is the region outside the disk of radius centered at the point . The sketch is as follows:

# 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,

# Use Taylor polynomials to approximate the nonzero root of arctan x = x2

1. Show that is an approximation of the nonzero root of the equation

using the cubic Taylor polynomial approximation to .

2. Given that

prove that the number from part (a) satisfies

Determine if is positive or negative and prove the result.

1. Proof. From a previous exercise (Section 7.8, Exercise #3) we know

So, to approximate the nonzero root of we have

2. We know from the same previous exercise we used in part (a) that the error term for satisfies the inequality

Using the values for and given we have

# Use Taylor polynomials to approximate the nonzero root of x2=sin x

1. Using the cubic Taylor polynomial approximation of , show that the nonzero root of the equation

is approximated by .

2. Using part (a) show that

given that . Determine whether is positive or negative and prove the result.

1. Proof. The cubic Taylor polynomial approximation of is

This implies

Therefore, we can approximate the nonzero root by

2. Proof. We know from this exercise (Section 7.8, Exercise #1) that for we have

So, for , and using the given inequality , we have

Furthermore, since

with the absolute value of each term in the sum strictly less than the absolute value of the previous term (since and ). Thus, each pair is positive, so the whole series is positive

# Prove an inequality for the error of the Taylor polynomial of arctan x

Prove that the error term in the Taylor expansion of satisfies the following inequality.

Proof. To prove this we will work directly from the definition of the error as an integral,

We know for we have, (we need for the expansion of to be valid),

Therefore we have

So, we can bound the error term by bounding the integral,

# Prove an inequality for the error of the Taylor polynomial of sin x

Prove that the error of the Taylor expansion of satisfies the following inequality.

Proof. Since the derivatives of are always , or we know that for we have . (In other words, the st derivative is bounded above by 1 and below by .) Therefore, we can apply Theorem 7.7 (p. 280 of Apostol) to estimate the error in Taylor’s formula at with and . For this gives us

Next, (from the second part of Theorem 7.7) if we have

# Prove an inequality of exponentials

For all and for any constants such that prove that

Proof. We want to consider the function

If we can show this function is decreasing on the positive real axis then we establish the inequality since this would mean that if then

(So, the trick here is to think of this as a function of the exponent. The and are some positive fixed constants.) To take the derivative of we use logarithmic differentiation,

Multiplying both sides by we then obtain

Now we can conclude that for all since the first term in the product

Since (any real power of a positive number is still positive) and . For the second term we have

since and are positive, but both logarithms are negative. We know these logarithms are negative since

implies

Hence, for all . This means is a decreasing function. Therefore, if then we have