Mathematical Preliminaries

Question 1

If the terms of an infinite series do not approach zero, can the series converge?

Answer 1

No

Question 2

If the terms of an infinite series approach zero, does that mean the series converges?

Answer 2

No

Question 3

What is the essence of the comparison test for series convergence?

Answer 3

If the terms of a series are always less than the terms of a convergent series, that series must be convergent as well. It’s basically the squeeze theorem.

Question 4

What is the essence of the Ratio test of series convergence?

Answer 4

If the ratio of one term in the sequence to the previous term approaches a value less than one, the series is convergent. If the ratio approaches a value greater than one, the series is divergent. If the ratio approaches one, the test is inconclusive.

Question 5

What is the essence of the integral test of series convergence?

Answer 5

If you can construct an integral which is greater than the sum of the terms of a series yet converges, then the series is convergent. Similarly, if you can construct an integral which is less than the sum of terms of a series yet diverges, the series is divergent.

Question 6

What is the Leibniz criterion regarding the convergence of an alternating series?

Answer 6

An alternating series is convergent if the magnitude of each summand decreases monotonically, approaching zero for large n.

Question 7

If you truncate an alternating series, the error will always be smaller than…

Answer 7

…the first term dropped from the series.

Question 8

What does it mean for a series to be absolutely convergent?

Answer 8

It means that the terms would still converge even if you took the absolute value of each one (as opposed to being conditionally convergent).

Question 9

When an alternating series converges but does so only because it is alternating, that series is said to be…

Answer 9

…conditionally convergent

Question 10

What is the practical significance of whether a series is absolutely or conditionally convergent?

Answer 10

A series that is absolutely convergent can have operations done on it and remain absolutely convergent. These operations include reordering the terms of the sum, and adding or multiplying it by another absolutely convergent series.

Question 11

Regarding the convergence of alternating series, what is known as “Riemann’s Theorem”?

Answer 11

A conditionally convergent series may be made to converge to any desired value or even to diverge, by changing the order in which the terms are summed.

Question 12

When facing a double series (a sum inside a sum), can you reverse the order of summation?

Answer 12

Yes

Question 13

What does it mean for a series (sum) of functions to be uniformly convergent?

Answer 13

It means that the rate of convergence of the series is not dependent on the argument of the functions (e.g. x)

Question 14

What is the most common test for uniform convergence of a series (sum) of functions and how does it work?

Answer 14

It is known as the Weierstrass M Test. The idea is that if you can construct a convergent series whose terms are always greater than the terms of the series of functions, no matter what x is (within a certain interval), then that series of functions must be uniformly convergent within that interval. The series must be absolutely convergent in order for the M test to work.

Question 15

What is the significance of having a uniformly convergent series of functions?

Answer 15

If a series is uniformly convergent, it can be integrated and differentiated term-by-term, and the sum will be continuous no matter what x (the argument of the functions) is.

Question 16

In general, what types of series are considered “Power Series”?

Answer 16

A series of the form f(x) = a₀+a₁x+a₂x²+a₃x³+…

is called a power series

Question 17

Give one example of how a power series is obtained.

Answer 17

A power series can be constructed by making a taylor expansion of a function.

Question 18

When replacing an expression with its power series/taylor expansion, what do you have to be mindful of?

Answer 18

How quickly or whether that series converges to the value that you need it to.

Question 19

Can there be multiple taylor expansions for a single function/expression?

Answer 19

No, each expression has a unique expansion

Question 20

What is L’hospital’s rule?

Answer 20

If the limit of the ratio of two functions can’t be found because it is indeterminate, you can take the limit of the ratio of the derivative of the respective functions instead.

Question 21

The binomial expansion is a power series for which class of expressions?

Answer 21

Expressions of the form (1+x)^m where x is between -1 and 1, and m can be anything.

Question 22

What is the essential idea of partial fractions decomposition?

Answer 22

It is like reverse process of finding the common denominator. You begin with one fraction that has a factorable denominator. Then you write it as a sum of multiple fractions, one with each factor as its denominator. Then you solve for the individual numerators.

Question 23

Which function is equal to this series?

Answer 23

e^x

Question 24

Which function is equal to this series?

Answer 24

sin(x)

Question 25

Which function is equal to this series?

Answer 25

cos(x)

Question 26

Which function is equal to this series?

Answer 26

cosh(x)

Question 27

Which function is equal to this series?

Answer 27

sinh(x)

Question 28

Which function is equal to this series?

Answer 28

ln(1 + x)

where x is between one and negative one

Question 29

Which function is equal to this series?

Answer 29

(1 - x)^-1

where x is beween one and negative one

Question 30

What does it mean for a set of vectors to span the spase?

Answer 30

It means that any vector in the space can be written as a linear combination of those vectors.

Question 31

What does it mean for a set of vectors to form a basis for a vector space?

Answer 31

It means that any vector can be written as a linear combination of that set of vectors, which are linearly independent from each other.

Question 32

What is the dimension of a vector space? (How do you find it?)

Answer 32

The dimension of a vector space is the minumum number of vectors required to span the space (or the number of vectors which form a basis for the space).

Question 33

What does the dot product of two vectors represent?

Answer 33

It tells you how much the two vectors point in the same direction.

Question 34

The dot product of two vectors is computed by…

Answer 34

…multiplying corresponding components and adding up those products.

Question 35

The dot product of a vector with itself gives…

Answer 35

…the square of the magnitude of the vector.

Question 36

What is the “geometric” formula for the dot product of two vectors?

Answer 36

Question 37

What does it mean for two vectors to be orthogonal?

Answer 37

It means that the dot product of the vectors is zero.

Question 38

Given a complex number z = x + iy, another complex number z* = x - iy, is called…

Answer 38

…the complex conjugate of z.

Question 39

In general, equations containing complex numbers are solved by…

Answer 39

…setting the real part part equal to the real part, and the imaginary part equal to the imaginary part.

Question 40

Is the space of complex numbers a vector space?

Answer 40

Yes.

Question 41

The product of a complex number with its conjugate gives…

Answer 41

…the square of the magnitude of the complex number.

Question 42

How do you divide two complex numbers?

Answer 42

Multiply the top and bottom of the fraction with the conjugate of the denominator, which will make the denominator real.

Question 43

In general, how do you plug a complex variable into a function?

Answer 43

Use the function’s power series, with a complex number z replacing x, and separate the real and imaginary parts of the series.

Question 44

Is Euler’s Formula valid for complex numbers as well?

Answer 44

Yes.

Question 45

How are the cartesian and polar representations of complex numbers related?

Answer 45

Question 46

What is the modulus of a complex number?

Answer 46

The magnitude or radius of the complex number from the origin.

Question 47

What is the argument of a complex number?

Answer 47

The angle or phase of the complex number, relative to the real axis.

Question 48

Visually, what does the product of two complex numbers look like?

Answer 48

The product will have a magnitude which is the product of the original magnitudes and a phase which is the sum of the original phases.

Question 49

Hyperbolic cosine is related to normal cosine through complex numbers. How?

Answer 49

cosh(iz) = cos(z)

Question 50

Hyperbolic sine is related to normal sign through complex numbers. How?

Answer 50

sinh(iz) = isin(z)

Question 51

What is Euler’s formula relating the exponential with the hyperbolic sine and cosine?

Answer 51

e^iz = cosh(iz) + sinh(iz)

Question 52

Since the trigonometric functions are related to the complex exponential, the inverse trigonometric functions are related to…

Answer 52

…the complex logarithm.

Question 53

How many values does the complex expression z^(1/n) have?

Answer 53

N values, each dividing the polar complex plane into equal angles.

Question 54

The logarithm of a complex number is complex in general and can be found using….

Answer 54

…the polar representation of the complex number, as well as logarithm rules.

Question 55

For some function f, how is df defined?

Answer 55

df is defined as how much the function changes as the argument changes by an amount dx. So df = (df/dx)dx. If f is a multivariable function, df is the sum of the previous expression for each variable.

Question 56

When taking the partial derivative of a function with respect to two or more variables, does it matter which order the partial derivatives are in?

Answer 56

No. Consecutive Partial derivatives can be computed in any order.

Question 57

What does this notation for a partial derivative mean?

Answer 57

It is identical to a normal partial derivative, just more explicit about y being held constant.

Question 58

What is the chain rule, generalized for multivariable functions?

Answer 58

Notice how the differentials cancel.

Question 59

What is a “stationary point” for some function f(x,y,z)

Answer 59

A point at which df/dx, df/dy, and df/dz are all zero. It can be either a maximum (if the second derivatives are negative in all directions) or a minimum (if the second derivatives are positive in all directions).

Question 60

In a multivariable function, what characterizes a “saddle point” ?

Answer 60

A stationary point at which the 2nd partial derivatives of the function are a mixture of positive and negative.

Question 61

How is the integration by parts formula derived?

Answer 61

Starting with the product rule, then integrating and rearranging terms.

Question 62

State the integration by parts formula in words.

Answer 62

The integral of u dv is equal to uv minus the integral of v du.

Question 63

How could Euler’s formula be helpful for solving this integral?

Answer 63

By realizing that cos(bt) is the same as the real part of e^ibt, you could integrate the product of two exponentials, which is really easy! (and then taking the real part at the end).

Question 64

What is “Integration by recursion”?

Answer 64

Sometimes, you can only find a recursive formula for an integral (e.g. when doing integration by parts). Then, the value of the initial integral may be recoverable using numerical or other means, so you can use the recursive formula to find all other integrals.

Question 65

What does it mean to evaluate a multidimensional integral using an “iterated integral”?

Answer 65

It means you integrate the variables in a sequence, starting with the innermost variable, and then integrating the results of that integral with respect to the next variable, and so on.

Question 66

When a single integral sign is used to represent a double integral, like in the image attached, how should the integral be interpreted?

Answer 66

An iterated integral, as usual. Just remember that the dA is a differential-area element, meaning it really stands for dxdy. Likewise, if there were a differential-volume element dV, it would stand for dxdydz. Although it can be thought of as the product of the individual differentials, an iterated integral is still appropriate.

Question 67

In a multiple-integral (double, triple, etc.), is it permissible to change the order of integration?

Answer 67

Yes, but you have to be careful to make sure that the region of integration that is represented stays the same.

Question 68

Why would you want to change the order of integration of a multiple integral?

Answer 68

It may allow you to avoid a tough integral.

Question 69

How do you decide which set of coordinates to use when evaluating an integral?

Answer 69

Use coordinates that match the symmetry of the problem, so that the functions you have to integrate are as simple as possible.

Question 70

In one-dimensional integrals, a change of variables (also known as a substitution or change of coordinates), requires two changes to the integral. What are they?

Answer 70

1. The differential element (dx) must be replaced by (dx/du)du (notice how the differentials cancel out to be equal to dx here)
2. The limits of the integration must be changed from x₁ and x₂ to u(x₁) and u(x₂)

Question 71

In one-dimensional integrals, a change in variables (also known as a substitution or change in coordinates) requires that the differential element dx is replaced by (dx/du)du. In multidimensional integrals, the differential element dxdy is replaced by…

Answer 71

Jdudv, where J is the jacobian matrix of the coordinate transformation, and du and dv are the differentials of the new coordinates. For example, in polar coordinates, the jacobian matrix is equal to r, so dxdy becomes rdrdø.