I prefer to explain the topic by emphasizing to students that they should not think of the limit of the series of a sequence as summation. The notation is very misleading, and it actually refers to something different than summation, even if it superficially may look like a generalization of summation when you are not thinking about the subject deeply. I prefer thinking of series as a kind of "transform" that you apply to a sequence. When you go from a sequence a(n) to its sequence of partial sums s(n), there is very specific "rule" involving summation that takes you from a to s, so it can be thought of as a "special transformation" in an informal sense. This transformation from a to s is very useful, which is why mathematicians care about it, and evaluating lim s (n -> ♾) is therefore also something we care about. Explaining it this way can help students build a healthy intuition on the subject without also being misled about what the symbols mean, but also without having to immediately bombard them with difficult-to-understand rigor. In fact, calculus in general is much healthier to learn intuitively as having limits, derivatives, series, and integerals, as kinds of useful "machines" or "transforms" that we apply to functions to accomplish different things, rather than doing the whole "slope and area" analogies in combination with "infinitesimals". The "machine"-based intuition not only is easier to make rigorous and easier to relate to, but it also generalizes much better to multivariable calculus and beyond, as opposed to the "slope"-based intuition, which only works for single-variable calculus and is still harder to grasp, even without the rigor baggage.
I remember that I first learned about series in my Calculus course, thought it was a super boring subject. After seeing them again in my Real Analysis course I then started to appreciate how interesting they are! Loved the video!
2:48 definition of infinite sum : limit of partial sum
I prefer to explain the topic by emphasizing to students that they should not think of the limit of the series of a sequence as summation. The notation is very misleading, and it actually refers to something different than summation, even if it superficially may look like a generalization of summation when you are not thinking about the subject deeply.
I prefer thinking of series as a kind of "transform" that you apply to a sequence. When you go from a sequence a(n) to its sequence of partial sums s(n), there is very specific "rule" involving summation that takes you from a to s, so it can be thought of as a "special transformation" in an informal sense. This transformation from a to s is very useful, which is why mathematicians care about it, and evaluating lim s (n -> ♾) is therefore also something we care about.
Explaining it this way can help students build a healthy intuition on the subject without also being misled about what the symbols mean, but also without having to immediately bombard them with difficult-to-understand rigor. In fact, calculus in general is much healthier to learn intuitively as having limits, derivatives, series, and integerals, as kinds of useful "machines" or "transforms" that we apply to functions to accomplish different things, rather than doing the whole "slope and area" analogies in combination with "infinitesimals". The "machine"-based intuition not only is easier to make rigorous and easier to relate to, but it also generalizes much better to multivariable calculus and beyond, as opposed to the "slope"-based intuition, which only works for single-variable calculus and is still harder to grasp, even without the rigor baggage.
I remember that I first learned about series in my Calculus course, thought it was a super boring subject. After seeing them again in my Real Analysis course I then started to appreciate how interesting they are! Loved the video!
Thanks a lot! Series are a very nice topic with a lot of connections into other fields. I will upload more videos about them soon :)
You had me worried with that first example
I hope that the worries have vanished later :)