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math104-s22:notes:lecture_4 [2022/01/22 09:57] pzhou created |
math104-s22:notes:lecture_4 [2022/01/27 10:46] (current) pzhou [Lecture 4] |
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* Cauchy sequence. | * Cauchy sequence. | ||
- | Discussion time: Ex 10.1, 10.6 in Ross | + | Discussion time: Ex 10.1, 10.7, 10.8 in Ross |
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+ | ==== limit goes to $+\infty$? ==== | ||
+ | This is in Definition 9.8. Informally, it says for any $M$ (no matter how large it is), the sequence will eventually stay above $M$. | ||
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+ | ==== Monotone Increasing Sequence ==== | ||
+ | A sequence that is (weakly) increasing, namely $a_{n+1} \geq a_n$. Examples: (a) $a_n = n$. (b) $a_n = 1 - n$. (c) $a_n = \log n$. | ||
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+ | **Theorem**: | ||
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+ | Just consider the increasing case, the decreasing case is similar. How do we construct such a limit? The limit is obviously the ' | ||
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+ | Let's look at the example 2 in Ross. Which is about recursive sequence | ||
+ | $$ s_n = \frac{s_{n-1}^2 + 5}{2 s_{n-1}} $$ | ||
+ | (such is an example of deterministic ' | ||
+ | Let's do the graphical trick. One draw the graph of $y = \frac{x^2+5}{2x}$ on the plane, and the diagonal $y = x$. Then, we can start from point $(x,y) = (s_1, s_1)$. Then, we | ||
+ | * go vertically to the graph, and we land on $(s_1, s_2)$ (since we go vertically, $x$ coordinate is fixed, and we move $y$ until we have $y = f(x)$) | ||
+ | * go horizontally to the diagonal, and we land on $(s_2, s_2)$ | ||
+ | * go vertically to the graph, and we land on $(s_2, s_3)$ | ||
+ | * .... | ||
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+ | This would show a zig-zag curve that goes to the intersection point of $y=x$ and $y=f(x)$, which is point $(s,s)$, where $s$ is the solution to equation $x=f(x)$. | ||
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+ | Now, to show that this converges, we apply the above theorem. Note, depending on initial condition, this will either be monotone increasing or decreasing. | ||
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+ | Unbounded monotone increasing (respectively, | ||
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+ | ==== $\liminf$ and $\limsup$ ==== | ||
+ | Recall the definition of $\sup$. For $S$ a subset of $\R$, bounded above, we define $\sup(S)$ to be the real number $a$, such that $a$ is $\geq$ than any element in $S$, and for any $\epsilon> | ||
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+ | Also, for a sequence $(a_n)_{n=m}^\infty$, | ||
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+ | Also, for a sequence $(a_n)_{n=1}^\infty$, | ||
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+ | We want to define a gadget, that captures the 'upper envelope' | ||
+ | $$ A_m = \sup_{n \geq m} a_n $$ | ||
+ | then we define | ||
+ | $$ \limsup a_n = \lim A_m (= \inf A_m) $$ | ||
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+ | Time for some examples, $a_n = (-1)^n (1/n)$. | ||
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