Summer of Math Exposition

Really well done! In addition to the nice visualizations and explanations, I want to specifically call out the excellent pacing: this is exactly the speed that is needed for undergraduate students learning this for the first time. The topic is built up from simpler ideas (sin x \approx x) and questions like “why should we do it this way?” are nicely answered. The examples are really nice for physics students, which is the stated target audience. I can see myself using this as part of a lecture or assigning it for students to watch on their own. Amazing work!

I really enjoyed this one. The explanations are great and every step that leads to the initial idea of the Taylor series is motivated very well. The visuals are very helpful, the voice is great to listen to! Just some small things: In the middle of the video it sounded like it is implied that the Taylor series will converge for points that are infinitely far away”. Neither is this true in general, demonstrated by the example only at the end of the video, nor is there an attempt to argue why the Taylor series should converge at all anywhere. But overall a great video!

This video does a solid job at explaining Taylor series, and I think a calculus student would find it very useful. However, most of the video focuses on something that would most likely be taught in class, including the visual intuition. Though the video was rather vanilla, the best part of the video is tucked at the end. When talking about the singularity, the explanation for radius of convergence was amazing and helped me understand radius of convergence better. If the video was about radius of convergence (and maybe the bit about dimensionless units), I think it would be more satisfying to watch. Good video, and I would love to see what you do in the future.

Very good job! I can see you put a lot of attention to detail in making the animations, so that every step can be easily understood.

The biggest issue might be novelty, since Taylor Series is a topic which has been broadly covered in many other videos, but you overcame this problem at the end of the video using a nice application to physics that fits perfectly. Also, I think the example you used to show the importance of the radius of convergence is great, as this is one of the aspects that many of the previous videos about Taylor Series fail to cover properly.

5:43: The aproximate = that appears should be =. That’s the definition of f1(x), there’s nothing aproximate about it, it’s axactly that. 8:51: What is “reasonably well-behaved”? 19:28: the whole section that ended here felt like it came out of thing air.

This was maybe a bit basic to be labelled “graduate,” but it was a really nice introduction to the idea of Taylor series.

I appreciate that you mention an example of a 0th order approximation. I also appreciate that you mention that the series only converges within some radius, and then connection with singularities (though you would need complex numbers to make it fully correct).

Yes! One of the videos I wanted to watch. Outstanding presentation

I think the video explains Taylor series very well, but I’m not sure whether it adds much on top of existing videos. I found the motivation with the pendulum rather weak. I believe it could have been more interesting if you had compared an animation of how the pendulum swings with the analytical approximation and a numerically exact solution. That could have visually quantified how well the approximation works. The example about gravity was executed well. I’m not sure whether adding the second solution strategy is confusing / detracting to your target audience. The note on what happens at singularities was well done, where the viewer gets a chance to spot it themselves before you point them towards it. In general, the pacing could have been faster for my taste. Not necessarily the speed of the voice over, but some of the pauses between sentences seemed too long to me. All in all, this is a good presentation of the topic. I just don’t see how it distinguishes itself from existing material.

I’ve always struggled understanding what’s behind Taylor series. Brilliant explanation!

Pretty nice explanation of Taylor series in the context of physics. The memorability part for me came at the end with the explanation of the radius of convergence being limited by reflections of singularities - I’m not sure I’ve seen that before. I also enjoyed the little graphic of the person standing on the globe. Overall, a very clear explanation that would be a good resource to a physics student.

Extremely high quality pedagogy here.

Immediately engaging, grounded in canonical/real-world examples from math/engineering, yet discusses these topics in familiar, casual language that makes the lesson very inviting to a broad viewership. Your pacing is excellent as well—you anticipate where the viewer may feel the need to pause to absorb information and include these recaps right in the video.

I especially loved your intuitive presentation on how to do better than straight lines and deriving power series coefficients. The cherry on top I would have loved to see added is a graphical sense of “We’ve made the approximation hug the function as best we can near the origin—how do we make it hug the approximation elsewhere?” This could have further eased the transition into using higher derivatives to generate coefficients of the approximation.

For the gravitational example, it would have been super cool to see a few other examples besides the ISS (say, top of a skyscraper or summit of Everest) to really solidify with real-world examples how good the constant approximation is near the surface. Field strength vs. distance would have also been a great plot to introduce before you made the problem dimensionless—possibly could tie it to the dwindling gravity experienced by a rocket on its way to the moon—and that way the general shape of the graph would be a familiar invariant for the viewer when you nondimensionalize the problem.

Also might be worth mentioning that the singularity is only a problem with point masses and briefly mention that gravity UNDER the Earth’s surface does not follow this same shape because the mass is distributed around the point of interest—a curious viewer might otherwise wonder why the center of the Earth doesn’t blow up! :D

Astonishingly strong/clear insights on the radius of convergence as it relates to evenness/oddness of Taylor terms and the long-distance affect of singularities, and a great way to very casually/naturally introduce the concept of small parameters. Bravo.

Thoroughly enjoyed this start to finish!

Great explanation. I liked the introduction of the radius of convergence via an example. Perhaps, one could trim the video to 15 minutes. One could also include an example when the taylor approximation fails (the classical example is f(x)=e^{-1/x^2})

I really enjoyed watching this video. The presentation quality was very high and I would say that it dealt with the subject matter well. My only gripe would be that since this video was labelled with the “graduate” tag, it should hint at the need for more mathematically rigourous arguments when speaking about issues of convergence. For example, for some functions like the exponential function, the convergence is uniform on compact sets, but in general, a smooth function isn’t guaranteed to equal its Taylor series. That’s where distinctions like pointwise vs. uniform convergence, compactness, and analyticity come in.

I’m not saying that this video should have been gone in depth with these concepts but maybe a comment along the lines of “while the Taylor series provides a good approximation for many functions you may come across some which won’t behave as nicely and require us to delve into more sophisticated notions of convergence which we’ll go into at another time”. Of course, there’s always a tradeoff between presenting a fully rigorous argument and crafting an engaging explanation for a broad audience: too much detail can obscure the main ideas, while simplifying for intuition makes the exposition clearer but risks omitting important caveats. I think this video does a good job with the presentation, pacing and keeping the viewer engaged. One could argue that it does lack a bit of novelty however as it does resembles certain aspects of this video: https://www.youtube.com/watch?v=3d6DsjIBzJ4