Day 36 in MAT229

1. A sequence is an infinite, ordered list of numbers: $\left\{a_k\right\}_{k=1}^{\infty}$

   We can think of it as a function $a(x)$, but with integers for arguments, and real numbers for values: \[ a: \mathbb{N} \rightarrow \mathbb{R} \]

   It might be better simply to write it as $a(n)$, but mathematicians have adopted the alternative notation $a_n$, perhaps to make clear that this function has a domain restricted to integers.

2. A series is an infinite sum of terms in a sequence. You can think of it as the analogue of an improper integral: \[ \sum_{k=1}^{\infty}a_k \]

   And this is one of the reasons that we are not too surprised when we find ourselves comparing series and improper integrals.

• I mentioned that $n!$ gives the number of ways of ordering $n$ distinct objects; that the factorial function is very useful in "combinatorial situations".
• I also mentioned, but wanted to emphasize, that it's really useful in Taylor polynomials: \[ T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k \] The factorials in this formula are there to "capture" the powers that come down when we differentiate; they assist in making sure that the Taylor polynomial and the function have their first $n$ derivatives in common at $x=a$.

  Better to look at a specific example: consider the cubic Taylor polynomial.

  \[ T_3(x) = \frac{f^{(0)}(a)}{0!} (x-a)^0 + \frac{f^{(1)}(a)}{1!} (x-a)^1 + \frac{f^{(2)}(a)}{2!} (x-a)^2 + \frac{f^{(3)}(a)}{3!} (x-a)^3 \] or \[ T_3(x) = f(a) + f'(a) (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \frac{f'''(a)}{3!} (x-a)^3 \] Then the derivative of $T_3(x)$ is \[ T_3'(x) = \sum_{k=1}^{3} \frac{f^{(k)}(a)}{(k-1)!} (x-a)^{k-1} \] or \[ T_3'(x) = f'(a) + \frac{f''(a)}{1!} (x-a)^1 + \frac{f'''(a)}{2!} (x-a)^2 \] Notice how we lost the $k=0$ term -- it was a constant, and when we differentiated we lost it. Continuing on, \[ T_3''(x) = f''(a) + \frac{f'''(a)}{1!} (x-a)^1 \] and \[ T_3'''(x) = f'''(a) \]

Remember that it's all about the tail. In terms of convergence, we don't care about any handful of terms at the head of a series: \[ \sum_{k=1}^\infty a_k = \sum_{k=1}^m a_k + \sum_{k=m+1}^\infty a_k \]

1. Comparison Tests (.nb)
2. Comparison Tests (.pdf)
3. Long talks you through the Comparison Test.

• Nothing to mention.