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# Determine the interval of convergence of ∑ xn / n! and show that it satisfies y′ = x + y

Consider the function defined by the power series

Determine the interval of convergence for and show that it satisfies the differential equation

(We might notice that this is almost the power series expansion for the exponential function and deduce the interval of convergence and the differential equation from properties of the exponential that we already know. We can do it from scratch just as easily though.)

First, we apply the ratio test

Hence, the series converges for all . Next, we take a derivative

Then we have

Now, to compute the sum we can solve the given differential equation

This is a first order linear differential equation of the form with and . We also know that ; therefore, this equation has a unique solution satisfying the given initial condition which is given by

Where and

Therefore, we have

# Prove some properties of complex numbers raised to complex powers

If is a nonzero complex numbers and let

where

1. Compute , , and .
2. Prove that if , and are in with .
3. What conditions on and must we have for the equation

to hold? Show that the equation fails when and .

1. The computations are as follows,

2. Proof. Using the definitions we compute,

3. First, if and then we have

but

Thus,

In order for we must have since

and only when .

# 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

# Prove DeMoivre’s theorem using complex numbers

1. Prove DeMoivre’s theorem,

for all and all .

2. Prove the triple angle formulas for sine and cosine,

by letting in part (a).

1. Proof. Since we have

2. Letting , we first apply DeMoivre’s theorem to get

On the other hand, we can expand the product,

Equating real and imaginary parts from the two expressions we obtain the requested identities:

# Prove formula relating trig functions of real numbers to the complex exponential

1. Prove that for we have the following formulas,

2. Using part (a) prove that

1. Proof. We compute, using the definition of the complex exponential, :

(Where in the second to last line we used that cosine is an even function and sine is odd, i.e., and .)
For the second formula we compute similarly,

2. Proof. We can compute these directly using the expressions we obtained in part (a),

# Prove that the complex exponential function is never zero

1. Prove that for all .
2. Find all such that .

1. Proof. Let for real numbers . Then from the definition of the complex exponential, and the fact that for complex numbers and from Theorem 9.3 (page 367 of Apostol) we have,

But from the properties of the real exponential function we know for any . Furthermore, we know for all since

which contradicts the Pythagorean identity (). Hence, for any

2. We compute

Then, setting real and imaginary parts equal this implies

This implies and . Thus,

# Compute some properties of compound interest rates

Consider a bank account which starts with dollars and pays an interest rate per year, compounding times per year.

1. Prove that the balance in the bank account at the end of years is

For fixed values of and , the balance at the end of years as is given by

We say that money grows at the annual rate with continuous compounding if the amount of money after years is denoted by is given by

Give an approximate length of time for the money in a bank account to double if and compounds:

1. continuously;
2. four times per year.

Incomplete. Sorry, I’ll try to get back to this soon(ish).

# Prove some properties of the function e-1/x2

Consider the function

and .

1. Prove that for every positive number we have

2. Prove that if then

where is a polynomial in .

3. Prove that

1. Proof. (A specific case of this general theorem is actually the first problem of this section, here. Maybe it’s worth taking a look since this proof is just generalizing that particular case.) We make the substitution , so that as and we have

by Theorem 7.11 (page 301 of Apostol) since implies as well

2. Proof. The proof is by induction on . In the case we have

So, indeed the formula is valid in the case . Assume then that the formula holds for some positive integer . We want to show this implies the formula holds for the case .

But then the leading term

is still a polynomial in since the derivative of a polynomial in is still a polynomial in , and so is the sum of two polynomials in . Therefore, we have that the formula holds for the case ; hence, it holds for all positive integers

3. Proof. The proof is by induction on . If then we use the limit definition of the derivative to compute the derivative at 0,

So, indeed and the statement is true for the case . Assume then that for some positive integer . Then, we use the limit definition of the derivative again to compute the derivative ,

This follow since is still a polynomial in , and by the definition of for . But then, by part (a) we know

Therefore,

Thus, the formula holds for the case , and hence, for all positive integers

# Find values for a constant such that a given limit will exist

Consider the limit expression

Find the value of the constant such that this limit will be finite. Find the value of the limit in this case.

We have

In order for this limit to exist we must have ; hence . The limit is then

# Find the limit as x goes to 0 of ((1/x) – (1/(ex – 1)))

Evaluate the limit.

We have