trying to be a mile wide AND a mile deep
Far from carefully vetted for accuracy. 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 […]
This is an exam problem I gave once; you’ll need to know |A| is the size/number of elements of A. Let A and B be finite nonempty sets, and f a function from A to B. Fill one I, for injective (1-1), and one S, for surjective (onto), in each line of the following table […]
Using the ease of moving between geometric series and their sums, plus the fact that for power series the derivative of a series is the sum of the derivatives of each term (and likewise for integrals), we can find a wider range of power series without too much trouble. I like making diagrams so here […]
Break into chunks and then add back together. Break into tasks and then multiply together. Do via the back door – find exactly when you don’t want and what’s left over is what you do want. Combining this with DeMorgan’s Laws can be powerful. Counting: find the total number of ways to accomplish a task […]
The key to the equations of lines and planes in three dimensions is that, in each case, we need a point to locate the object in space, and a vector to tilt it at the correct angle. In each case, however, the kind of vector that unambiguously gives the direction of the object is different. […]
To work with factorial in series, you often simply need the ratio test. However, sometimes the ratio test doesn’t give a tractable fraction. In those cases it is good to remember the definition of factorial, in particular the fact that n! = n·(n-1)!, and that tests for convergence typically have conditions that need only hold […]