trying to be a mile wide AND a mile deep
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 […]
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 […]
In a number of tests for series convergence and divergence, you locate or calculate a quantity and draw conclusions based on its value. Here’s a table of which values give what conclusions, for five such tests. Note that the table assumes the series is of the correct form for the test to apply at all […]
There are basically 4 techniques for solving an indefinite integral. Directly/by rules. After algebraic manipulation. Using substitution. By parts. Using limits for improper bounds or internal discontinuities should be mentioned here, but it’s not really in the same category since you don’t use the limit to find an antiderivative. is the method you learn first. […]
I learned this visualization of the chain rule from one of my grad school officemates. It draws on the fact that “and” usually goes with multiplication (more on that in its own post eventually). Imagine when you have f(g(x)) that x is a little sealed box. It is inside a second sealed box marked g, […]