Day 38 in MAT229

Today:

• Announcements:
  • Weekly Assignment 6 is ... not graded yet. Rats.

  • Weekly #7 is not due until Tuesday (it was originally Friday on the schedule, but I'm trying to give you a week or so for these homeworks).

  • Early head's up: exam 3 is Tuesday, 4/9.

  • James found an error on part a, Q3, 11.6. If anyone else had a problem, let me know.

• Today:
  • We are wrapping up Sect. 11.8: Power series (pdf, pdf summary) which takes us in a different, and very important, direction:
    What if we replace the constants in the series with functions?

    In particular, we focus on an infinite sum of increasing powers of ${\displaystyle x}$ -- a polynomial of "infinite degree", essentially.

    Last time, at the buzzer, we discovered that the tremendously important exponential function \[ e^x=\sum_{k=0}^\infty\frac{x^k}{k!} \]

    We'll see that most other functions we deal with can be expressed this way, and it's key for computing approximations to these functions (along with the tools we learned about infinite series -- because when you replace ${\displaystyle x}$ with a value, we just have an infinite series, which we can approximate if it's convergent).

  • So it's on to Sect. 11.9: Functions as power series (pdf, pdf summary)