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math104-f21:midterm1-review [2021/09/17 21:05] pzhou [Midterm 1: Review] |
math104-f21:midterm1-review [2022/01/11 08:36] (current) pzhou ↷ Page moved from math104:midterm1-review to math104-f21:midterm1-review |
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- | ==== Sample Problems ==== | + | ===== Sample Problems |
+ | < | ||
+ | |||
+ | === True or False? If true, prove your result; if false, give a counter example. === | ||
+ | - Let $(a_n)$ be a Cauchy sequence of irrational numbers, then its limit has to be an irrational number. | ||
+ | - Let $S$ be an ordered set, then any non-empty finite subset $E \In S$ has a least upper bound. | ||
+ | - If $x$ is a limit point of sequence $(a_n)$, then there exists one $n \in \N$, such that $a_n = x$. | ||
+ | - If $(a_n)$ is a sequence bounded above, and $L = \limsup(a_n)$, | ||
+ | - Let $(a_n)$ be a bounded sequence in $\R$. Let $A_n = \sup \{a_m : 0 \leq m \leq n \}$, then $\lim A_n = \limsup a_n$ | ||
+ | - Let $(a_n)$ and $(b_n)$ be convergent sequences with the same limit $x$. And choose any function $f: \N \to \{0,1\}$. We define a new sequence, by mixing $a_n$ and $b_n$ $$c_n = \begin{cases} a_n & \text{if } f(n) = 0 \cr b_n & \text{ if } f(n) = 1 \end{cases} $$ Then, $c_n$ converges to $x$. | ||
+ | - If $(a_n)$ and $(b_n)$ are Cauchy sequence in $\R$, and they satisfy that $\lim(a_n b_n) = 1$, then $\lim a_n \neq 0$. | ||
+ | - If $(a_n)$ is a sequence of positive real numbers, for $n \geq 1$, and $A_n = (a_1 + \cdots + a_n) / n$, show that if $a_n$ is convergent then $A_n$ is convergent. Give an example where $A_n$ is convergent, but $a_n$ is not convergent. | ||
+ | - Is there a sequence $(a_n)$, where $|a_n - a_{n-1}|$ is monotone decreasing, but $(a_n)$ is not convergent? | ||
+ | |||
+ | ===== Solution ===== | ||
+ | ==== 1 ==== | ||
+ | Let $(a_n)$ be a Cauchy sequence of irrational numbers, then its limit has to be an irrational number. | ||
+ | |||
+ | False. Say $a_n = \pi / n$. | ||
+ | |||
+ | ==== 2 ==== | ||
+ | Let $S$ be an ordered set, then any non-empty finite subset $E \In S$ has a least upper bound. | ||
+ | |||
+ | True. One can prove this by induction on the size of $E$. | ||
+ | |||
+ | ==== 3 ==== | ||
+ | If $x$ is a limit point of sequence $(a_n)$, then there exists one $n \in \N$, such that $a_n = x$. | ||
+ | |||
+ | False. $0 = \lim_n 1/n$. | ||
+ | |||
+ | ==== 4 ==== | ||
+ | If $(a_n)$ is a sequence bounded above, and $L = \limsup(a_n)$, | ||
+ | |||
+ | True. explained in class. | ||
+ | |||
+ | ==== 5 ==== | ||
+ | Let $(a_n)$ be a bounded sequence in $\R$. Let $A_n = \sup \{a_m : 0 \leq m \leq n \}$, then $\lim A_n = \limsup a_n$. | ||
+ | |||
+ | False. That's not the definition of $\limsup$. Say $a_n = 1/(n+1)$, for $n \in \N$, then $\lim A_n = 1$, and $\limsup a_n=0$. | ||
+ | |||
+ | ==== 6 ==== | ||
+ | Let $(a_n)$ and $(b_n)$ be convergent sequences with the same limit $x$. And choose any function $f: \N \to \{0,1\}$. We define a new sequence, by mixing $a_n$ and $b_n$ $$c_n = \begin{cases} a_n & \text{if } f(n) = 0 \cr b_n & \text{ if } f(n) = 1 \end{cases} $$ Then, $c_n$ converges to $x$. | ||
+ | |||
+ | True. By convergence of $a_n$ and $b_n$, for any $\epsilon> | ||
+ | |||
+ | ==== 7 ==== | ||
+ | If $(a_n)$ and $(b_n)$ are Cauchy sequence in $\R$, and they satisfy that $\lim(a_n b_n) = 1$, then $\lim a_n \neq 0$. | ||
+ | |||
+ | True. Since if $a_n, b_n$ are convergent, then $1 = \lim(a_n b_n) = (\lim a_n ) (\lim b_n)$, hence $\lim a_n \neq 0$. | ||
+ | |||
+ | |||
+ | ==== 8 ==== | ||
+ | If $(a_n)$ is a sequence of positive real numbers, for $n \geq 1$, and $A_n = (a_1 + \cdots + a_n) / n$, show that if $a_n$ is convergent then $A_n$ is convergent. Give an example where $A_n$ is convergent, but $a_n$ is not convergent. | ||
+ | |||
+ | Proof: Let $x$ be the limit of $a_n$. Given any $\epsilon > 0$, we claim there exists an $N>0$, such that $|A_n - x| < \epsilon$ for all $n > N$. Let $\epsilon_1 = \epsilon/ | ||
+ | $$ A_n - x = (1/n) \sum_{m=1}^{N_1} a_m + (1/n) \sum_{m=N_1+1}^n (a_m-x) + (N_1/n) x $$ | ||
+ | Thus, we have | ||
+ | $$ |A_n - x| \leq (1/n) N_1 M + (1 - N_1/n) \epsilon_1 + (N_1/n) x $$ | ||
+ | we may choose $N$ large enough, such that $N_1 / N x < \epsilon/ | ||
+ | $$ |A_n - x| \leq \epsilon/3 + \epsilon / 3 + \epsilon / 3 = \epsilon. $$ | ||
+ | This finishes the proof of the first claim. | ||
+ | |||
+ | Here is an example where $A_n$ is convergent, but $a_n$ is not convergent: $a_n = (-1)^n$. | ||
+ | |||
+ | |||
+ | ==== 9 ==== | ||
+ | Is there a sequence $(a_n)$, where $|a_n - a_{n-1}|$ is monotone decreasing, but $(a_n)$ is not convergent? | ||
+ | |||
+ | Yes, for $n \geq 1$, $a_n = \sum_{m=1}^n 1/n$. |