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 […]
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 […]
Happy St. Patrick’s Day! Here, have an awkward flow chart.
1. “Sufficiently large” is an important concept in sequences and series. In essence it means any crazy thing can happen for as long as it wants to happen, so long as there is a finite point after which the sequence or series starts behaving in a controlled/predictable way. A finite number of terms can’t affect […]
I like to present totally cooked-up examples demonstrating the reason for certain hypotheses or limitations in calculus, in the hope that they will help students remember and correctly apply the theorems more easily. Here are a few. Note that the sequence has limit 0 but the function oscillates. This is why you can use convergence […]