My 2 developing from a typewritten style to one with a loop at the bottom left may have preceded mathematics. I just like the look. However, it probably would have happened eventually anyway – I cross my Zs for good measure, so they are decidedly not 2s.

I also cross my 7s, and in many cases (though not typically when it is part of a longer number) I give my 1 a top hook and a bottom bar. My lowercase i and t curve up at the bottom right, and my lowercase x has a hook at the upper left.

I’ve been out of mathematics for nearly two and a half years now, but those handwriting changes show no sign of fading: an odd little calling card left by my former vocation.

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Inclusive or is a low hurdle. In plain English, when someone asks “Should we take our vacation to New York or Boston?” the assumption is that the answer will be “New York” or “Boston” (or “I don’t care” or “neither”). The geeky joke – sometimes serious – answer of “yes” is totally unhelpful. However, it’s not too hard to get used to inclusive or, and we do have examples in natural language. One of the best is “Would you like sugar or cream in your coffee?” Of course, even then “yes” isn’t a useful answer, since there are three possible coffee fixings that would lead to it.

Implication is a much higher bar; there’s still a part of me, even, that doesn’t think A implies B means much of anything when A is false. Implications where there is clearly no causal relationship between A and B can be helpful, since they rarely appear in plain English (outside of statistical correlations, I suppose) and thus resist natural language intuition. For teaching, in addition to that, I settled on the approach of “an implication is true unless proven false.” You can only prove that it’s false by having A be true and B be false, so in the A-false situations the implication is therefore true. This is basically the conversion of “A implies B” into the disjunction “B or not-A,” but hopefully in a way that doesn’t just shift the confusion to a different location.

I think the base of that confusion is a disconnect between allowed truth values. In plain English, a sentence can be true, false, or nonsensical. The third option is not permitted in mathematics (except in the sense of ill-formed formulas), and it is confusing that many implications that seem nonsensical or are constructed from false clauses (“if the moon is made of green cheese, then fish swim in the sea;” “if the moon is made of green cheese, then carriages turn into pumpkins at midnight”) are logically true.

In the fifteen-plus years since my bridge class, I have found only one plain English example of an implication considered true but constructed with false clauses, and in general my students were unfamiliar with it: the adage “If wishes were horses, beggars would ride.” I would love another, even though nowadays it would be purely for my own interest.

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We must be careful not to confuse data with the abstractions we use to analyze them.
William James

A judicious man looks on statistics not to get knowledge, but to save himself from having ignorance foisted on him.
Thomas Carlyle

After all, facts are facts, and although we may quote one to another with a chuckle the words of the Wise Statesman, “Lies – damned lies – and statistics,” still there are some easy figures the simplest must understand, and the astutest cannot wriggle out of.
Leonard Courtney

The general who wins the battle makes many calculations in his temple before the battle is fought. The general who loses makes but few calculations beforehand.
Sun Tzu

Like the ski resort of girls looking for husbands and husbands looking for girls, the situation is not as symmetrical as it might seem.
Alan McKay (via the unix program fortune, as I recall)

The final mystery is oneself… Who can calculate the orbit of his own soul?
Oscar Wilde

He uses statistics as a drunken man uses lamp-posts… for support rather than illumination.
Andrew Lang

How dare we speak of the laws of chance? Is not chance the antithesis of all law?
Joseph Bertrand

What would life be without arithmetic, but a scene of horrors?
Rev. Sydney Smith

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e) Working only from the definitions of function, surjectivity, and injectivity, and not from other prior results, prove your answer for either line c) or line d) above. If there is an entry in the middle column, pin down the conditions under which that condition will hold of f.

No answers on this one, but a hint for part e): think of the function as a relation. That’s often helpful in working with functions between finite sets.

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1. Find a power series representation of

This function differentiates to a fraction we can interpret as the sum of a geometric series.

To complete Step 4, note that for x=0, = 0 and the series also equals 0, so C must be 0. The radius of convergence for the series obtained in Step 2 is 1, with an interval of (-1, 1). When you integrate the interval of convergence can gain or lose endpoints, but that’s a small thing to check after this fairly quick conversion.

2. Find a power series representation of

Instead of differentiating, this time we integrate to get the sum of a geometric series.

In Step 3 note that from the perspective of the derivative operator is constant. There’s no constant of integration to find so once you’ve rounded the bases you’re done – well, except for thinking about the radius and interval of convergence.

One more after the jump.

3. Find a power series representation for

This time we differentiate twice.

Once again you do not have to find C at the end, but in this case it is because C stays undefined: the series is intended to equal an indefinite integral, not a specific function. C does not affect the radius or interval of convergence of the series.

We can use the series above to find a power series for the definite integral Since plugging 0 into the series obtained in Step 6 gives 0, the answer is the result of plugging in 1:

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For a line, there is only one way to be parallel, but infinitely many ways to be orthogonal (think: every vector parallel to the xy-plane is orthogonal to the z-axis). Therefore the vector we want, the direction vector, is parallel to the line.

For a plane, conversely, there are infinitely many ways to be parallel, but only one way to be orthogonal (any vector orthogonal to the xy-plane is parallel to the z-axis). Therefore the vector we want, the normal vector, is orthogonal to the plane.

You may obtain these two pieces in many ways. In addition to being given the point and direction vector immediately, the following are enough to determine a line:

• two points on the line
• a point and a parallel line
• a point and two nonparallel vectors orthogonal to the line
• a point and two nonparallel lines orthogonal to the desired line
• a point and an orthogonal plane
• two intersecting (nonidentical) planes (the line of intersection)
• two intersecting (nonidentical) lines (the line through their point of intersection and orthogonal to both)

The following are enough to define a plane, in addition to being given a point and normal vector directly:

• three points in the plane
• a point and an orthogonal line
• a point and a line in the plane not containing that point
• two lines in the plane
• a point and a parallel plane
• a point and two planes orthogonal to the desired plane

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1. For example, the sum from n=1 to infinity of (n!)/(n^n). Once n is at least 2, we can compare to a nice series we know about.

The sum from n=1 to infinity of 2/(n^2) is easily shown to converge, and, from a finite point on, it sits on top of our original series, which has only positive terms and hence must also converge.

2. We could use the ratio test for the sum , but we can also use the alternating series test. A similar trick to the previous example shows the magnitudes of the terms have limit 0. Once n is at least 3, we have the following comparison on the absolute value of each term.

That last fraction has a limit of 0 as n goes to infinity.

However, for the alternating series test we need more than a limit of zero; the magnitudes must decrease to 0, for which we use the fact that (n+1)! can be written in terms of n!:

As long as n is at least 3, the n+1st term is the nth term times a value less than 1. The series doesn’t decrease right from the start, but from a finite point on it always decreases, and that is enough.

3. How about the sum from n=1 to infinity of ? You might be tempted to use the ratio test for this example, but it would waste your time. The terms simplify to [(n+1)(n+2)]/n^2, and have a limit of 1 as n goes to infinity, and therefore the series diverges by the test for divergence.

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Test	Value to find	Convergent	Divergent	Inconclusive
test for divergence		N/A	≠ 0	= 0
p-series	p in	> 1	≤ 1	N/A
geometric series	|r| in	< 1	≥ 1	N/A
ratio test		< 1	> 1	= 1
root test		< 1	> 1	= 1

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2. Whenever we want to isolate the direction of something, we use a unit vector.

3. While it is useful to think of linearly independent sets as “small” and spanning sets as “large”, be careful not to try to apply the converse. {0} is very small, but linearly dependent. {v, 2v, 3v, 4v, 5v, …} is very large, but spans only a dimension 1 space.

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