August 30: Review of Calculus

today we will go over sequence of numbers and limit.

Let $a_1, a_2, \cdots $ be a sequence of numbers. We can have many examples of it.

• $1,-1,1,-1\cdots $
• $0.9,0.99, 0.999, $
• $1,2,3, \cdots, $

We say a sequence $(a_n)$ converges to $a$, if for any $\epsilon>0$, there exists $N>0$, such that for any $n > N$, we have $|a_n - a| \leq \epsilon$.

a series is something that looks like $\sum_{n=1}^\infty a_n$. We can define the partial sum $S_n = \sum_{j=1}^n a_j$. We say the series $\sum_n a_n$ convergers if and only the partial sum converges.

Series is like a discretized version of integral.

1. what does absolute convergence mean for series?

2. the model convergent series

• $\sum_n 1/n^p$ for $p > 1$.
• $\sum_n 1/r^n$ for $r>1$

3. various tests

• comparison test, if $0<a_n < b_n$, and $\sum_n b_n$ converges, then $\sum_n a_n$ converges.
• ratio test
• root test

some exercise from Boas's textbook, try 1,3, 4-8