Back when I really didn’t know what I was doing, I tried to memorize a textbook verbatim. Fortunately, that didn’t last long. Even with good technique, memorization is effortful and time-consuming. To get the most benefit, we need to do it efficiently.
What does that mean?
Efficient memorization is building the minimal memory that lets you construct the result.
Let’s expand on that with an example.
Taylor’s Inequality is a key theorem related to Taylor Series, a powerful tool used widely across science and engineering. Taylor’s Inequality gives us a way of showing that a function is or is not equal to its Taylor Series representation.
Briefly, Taylor’s Inequality starts with an inequality:
If true on a given interval, then Taylor’s Inequality states that on that same interval,
My textbook doesn’t give a proof, just 8 steps of calculations that show this is true for n = 1, and saying that the same process will show this theorem holds for any other choice of n.
If you launched straight into trying to remember these steps in the proof, you’d exhaust yourself and gain little insight.
It pays huge dividends to notice they’re just taking that first inequality, integrating it on the interval (a, x) n+1 times (with some algebraic rearrangements in between), and then using the definition of Rn to relabel the result.
The result looks intimidating. It’s less so if you notice that the right hand side is just the absolute value of the n+1 term in the Taylor series, but replacing fn+1(a) with M.
Once you’ve identified these ways of compressing the information, you can memorize the key points much more easily and will be able, with enough time, to reconstruct the textbook’s illustration with pencil and paper.
How do I incorporate this into my memory “palace” for power series?
I visualize the symbol f(x) with bands of light above and below it, representing its integrals and derivatives.
I pick out one of the ones below it, representing the n+1th derivative, and set it up inside absolute value brackets as part of the inequality.
I picture what I know is the general form of the Taylor series next to it, and put it in the inequality with |Rn(x)| from the result. Just like you can focus on one part of a sculpture while ignoring the rest, I focus on the fn(a) part and ignore the rest. I pick up the M from the inequality, and stick it into the Taylor series “sculpture” to replace the fn(a) term with M.
I imagine sticking n+1 integral symbols in front of the first inequality, to represent what I have to do to reform it into the second equation.
The mathematical definition of Rn is stored elsewhere, so I drag it over to remind myself that I’m going to eventually have to rearrange my integrations so that they can be replaced by Rn
Back when I really didn’t know what I was doing, I tried to memorize a textbook verbatim. Fortunately, that didn’t last long. Even with good technique, memorization is effortful and time-consuming. To get the most benefit, we need to do it efficiently.
What does that mean?
Efficient memorization is building the minimal memory that lets you construct the result.
Let’s expand on that with an example.
Taylor’s Inequality is a key theorem related to Taylor Series, a powerful tool used widely across science and engineering. Taylor’s Inequality gives us a way of showing that a function is or is not equal to its Taylor Series representation.
Briefly, Taylor’s Inequality starts with an inequality:
If true on a given interval, then Taylor’s Inequality states that on that same interval,
My textbook doesn’t give a proof, just 8 steps of calculations that show this is true for n = 1, and saying that the same process will show this theorem holds for any other choice of n.
If you launched straight into trying to remember these steps in the proof, you’d exhaust yourself and gain little insight.
It pays huge dividends to notice they’re just taking that first inequality, integrating it on the interval (a, x) n+1 times (with some algebraic rearrangements in between), and then using the definition of Rn to relabel the result.
The result looks intimidating. It’s less so if you notice that the right hand side is just the absolute value of the n+1 term in the Taylor series, but replacing fn+1(a) with M.
Once you’ve identified these ways of compressing the information, you can memorize the key points much more easily and will be able, with enough time, to reconstruct the textbook’s illustration with pencil and paper.
How do I incorporate this into my memory “palace” for power series?
I visualize the symbol f(x) with bands of light above and below it, representing its integrals and derivatives.
I pick out one of the ones below it, representing the n+1th derivative, and set it up inside absolute value brackets as part of the inequality.
I picture what I know is the general form of the Taylor series next to it, and put it in the inequality with |Rn(x)| from the result. Just like you can focus on one part of a sculpture while ignoring the rest, I focus on the fn(a) part and ignore the rest. I pick up the M from the inequality, and stick it into the Taylor series “sculpture” to replace the fn(a) term with M.
I imagine sticking n+1 integral symbols in front of the first inequality, to represent what I have to do to reform it into the second equation.
The mathematical definition of Rn is stored elsewhere, so I drag it over to remind myself that I’m going to eventually have to rearrange my integrations so that they can be replaced by Rn