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# Establish the given limit relations

Use the previous exercise (Section 10.4, Exercise #34) to establish each of the following limits.

1. .
2. .
3. .
4. .
5. .
6. .

1. Let , then from Exercise #34 we know

where

Thus,

(since and then use the squeeze theorem). So,

2. Let

So,

Thus,

3. Let

Thus,

Therefore,

4. Let

Thus,

So,

5. Let

Thus,

Therefore,

6. Let

Thus,

Therefore,

# Prove that sums of trig functions can be expressed as sums of complex exponentials

Prove that sums of the form

are equal to sums of the form

Proof. We compute this directly, substituting in the formulas for sine and cosine in terms of the complex exponential,

that we derived in this exercise. So, we have

where

# Find the limit of the given function

Find the value of the following limit.

First, we use the algebraic identity,

to obtain

Then, we apply L’Hopital’s rule twice,

# Find the Taylor polynomial of log ((1+x)/(1-x))1/2

Show that

First, we have

We know from the previous exercise (Section 7.4, Exercise #6) that

We also know (from the example on page 277) that

Therefore, using Theorem 7.2 (a), the linearity property of we have

However, if is even and if is odd. Therefore we can sum over just the odd values of . Let and we have

Where we have renamed the index of summation in the final step so that the sum is over as in the book.

# Find the derivative of a product of terms (x-ai)bi

Find the derivative of the following function:

To take this derivative we want to use logarithmic differentiation. To that end we take the derivative of both sides,

Therefore, taking the derivative of both sides, we have

# Prove inequalities of the logarithm with respect to some series

Consider a partition of the interval for some .

1. Find step functions that are constant on the open subintervals of and integrate to derive the inequalities:

2. Give a geometric interpretation of the inequalities in part (a).
3. Find a particular partition (i.e., choose particular values for ) to establish the following inequalities for ,

1. Proof. We define step function and by

Since is strictly decreasing on , we have

Therefore, using the definition of the integral of a step function as a sum,

2. Geometrically, these inequalities say that the area under the curve lies between the step functions that take on the values and for each .
3. Proof. To establish these inequalities we pick the partition,

Then, applying part (a) we have

The final line follows since so the sum on the left starts with and the sum on the right only runs to . These were the inequalities requested

# Determine formulas for given sums

Consider the formula

By differentiating, determine formulas for the following:

1. .
2. .

1. We observe that

Thus, using the given formula we have,

2. (Note: There’s an alternate solution to this in the comments that is more direct than this one. Definitely worth taking a look at that alternative.)
Taking derivatives of both sides of the equation we derived in part (a),

Then, looking at the second term in the sum on the left, we have,

Thus, plugging this back into the expression above,

3. ( Note: This is a lot of algebra. I included as many steps as I thought appropriate, but that still has a lot going on in each step. If there is something that isn’t clear please leave a comment. Also, given all of these equations it seems likely I’ve made at least one typo / mathematical error. If you find one, please point it out.)

# Prove a formula for the sum from k=1 to n of sin kx

Prove

for .

Proof. First, we recall from this exercise

Adding, with the right side telescoping, as in the previous exercise, we have

# Prove a formula for the sum from k=1 to n of cos kx

Given the identity

prove

for .

Proof. Starting with the given identity we sum over ,

# More integrals of step functions

1. Compute

2. For prove

1. We compute,

2. Proof. Let . Then is a partition of and is constant on the open subintervals of . Further, for we have . Thus, we have (using this exercise and this exercise to evaluate some of the sums),