better frizzle

_posts/2024-05-08-approximating-pi.md

@@ -132,9 +132,9 @@ I guess we're still stuck in black-and-white infomercial territory:
 
 Before we try anything else, we should understand the problem:
 
-![Ms. Frizzle saying "if at first you don't succeed..."](/images/frizzle-1.jpg){width=50%} ![Ms. Frizzle saying "... find out why"](/images/frizzle-2.jpg){width=50%}
+![Ms. Frizzle saying "if at first you don't succeed, find out why"](/images/frizzle.png)
 
-I wasn't just talking about convergence behavior above for reasons of nitpickery (although I am not immune to such impulses). This is actually the precise reason why this series converges so slowly: we are as far away from the center of convergence as it is possible to be and still converge. From this lens, *of course* we're converging super slow: we have some sense that the goodness of an approximation is controlled both by the degree of the approximation and by the distance away from the center.
+I wasn't just talking about convergence behavior above for reasons of nitpickery (although I am not immune to such impulses). This is actually the precise reason why this series converges so slowly. Remember that our interval of convergence was $-1 \leq x \leq 1$, and we are in fact dealing with $x=1$. Therefore, we are as far away from the center of convergence as it is possible to be and still converge. From this lens, *of course* we're converging super slow: we have some sense that the goodness of an approximation is controlled both by the degree of the approximation and by the distance away from the center.
 
 This also gives me an idea for how to improve this process for approximating π: maybe there's a better point, further inside the interval of convergence, where we'll get faster convergence behavior. Hmm...