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- Real Analysis: Short Questions and MCQs @msc:mcqs_short_questions
- of real numbers countable? - Give an example of sequence, which is bounded but not convergent. - Is every bounded sequence convergent? - Is product of two convergent sequ... hat $\left\{\frac{1}{n+1} \right\}$ is decreasing sequence. - Is the sequence $\left\{\frac{n+2}{n+1} \right\}$ is increasing or decreasing? - If the sequence $\{
- MTH321: Real Analysis I (Spring 2020) @atiq
- on. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems ... . * Numerical Sequences and Series. Limit of a Sequence. Bounded Sequences. Monotone Sequences. Limits Su... e Questions==== ===Chapter 02=== * 2.01- Define sequence of real numbers. * 2.02- Define subsequence * 2.03- Define increasing sequence. * 2.04- Define decreasing sequence. * 2.05-
- Real Analysis Notes by Prof Syed Gul Shah @notes
- * Chapter 01: Real Number system * Chapter 02: Sequence and Series * Sequence, Subsequence, Increasing Sequence, Decreasing Sequence, Monotonic Sequence, Strictly Increasing or Decreasing * Bernoulli’s Inequality
- Chapter 02 - Sequence and Series @msc:real_analysis_notes_by_syed_gul_shah
- ====== Chapter 02 - Sequence and Series ====== ==== Contents ==== * Sequence, Subsequence, Increasing Sequence, Decreasing Sequence, Monotonic Sequence, Strictly Increasing or Decreasing * Bernoulli’s Inequality *
- MTH321: Real Analysis I (Fall 2021) @atiq
- on. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems ... . * Numerical Sequences and Series. Limit of a Sequence. Bounded Sequences. Monotone Sequences. Limits Su... at $x=\sup E$. ===Chapter 02=== * 2.01- Define sequence of real numbers. * 2.02- Define subsequence * 2.03- Define increasing sequence. * 2.04- Define decreasing sequence. * 2.05-
- Definitions: FSc Part 1 (Mathematics): PTB @fsc-part1-ptb
- . $2x=3$, if $x=\frac{2}{3}$. ===== Chapter 06: Sequence and series ===== * **Sequence:** Sequence is a function whose domain is subset of the set of natural numbers. * **Real sequence:** If all members of a sequence are real numbers,
- MTH322: Real Analysis II (Fall 2021) @atiq
- Chapter 02** - Define pointwise convergence of sequence of function. - Define uniform convergence of sequence of function. - Define pointwise convergence of se... ove Cauchy’s criterion for uniform convergence of sequence of functions. - State and prove Cauchy’s criter... onvergence of series of functions. - Consider a sequence of function $\{f_n\}$, where $f_n(x)=\frac{nx}{1+
- MTH321: Real Analysis I (Spring 2023) @atiq
- on. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems ... == **Questions from Chapter 02** - A convergent sequence of real number has one and only one limit (i.e. limit of the sequence is unique.) - Prove that every convergent sequence is bounded. - Suppose that $\left\{ {{s}_{n}} \right
- MCQs or Short Questions @atiq:sp15-mth321
- * (C) does not exist * (D) 0 - A convergent sequence has only ................ limit(s). * (A) one... two * (C) three * (D) None of these - A sequence $\{s_n\}$ is said to be bounded if * (A) ther... r all $n\in\mathbb{Z}$. * (D) the term of the sequence lies in a vertical strip of finite width. - If the sequence is convergent then * (A) it has two limits.
- Question 3 and 4 Exercise 4.1 @math-11-kpk:sol:unit04
- s of Question 3 and 4 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... =Question 3(i)==== Write down the nth term of the sequence as suggested by the pattern. $\dfrac{1}{2}, \dfra... \ldots$ ====Solution==== We can reform the given sequence to pick the pattern of the sequence as: $$\dfrac{1}{1+1}, \dfrac{2}{2+1}, \dfrac{3}{3+1}, \dfrac{4}{4+1},..
- Question 1 and 2 Exercise 4.1 @math-11-kpk:sol:unit04
- s of Question 1 and 2 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathema... 4,6,8, \ldots ,50$ ====Solution==== It is finite sequence whose last term is $50 $. =====Question 1(ii)===... 0,1,0,1, \ldots$. ====Solution==== It is infinite sequence, the last term may be $0$ or $1$ , but how much terms in this sequence, we don't know. =====Question 1(iii)===== Classi
- MTH322: Real Analysis II (Spring 2023) @atiq
- convergent. **Questions from Chapter 02:** - A sequence of functions $\{f_n\}$ defined on $[a,b]$ converg... \hbox{ and } x\in [a,b].$$ - Let $\{f_n\}$ be a sequence of functions, such that $\lim\limits_{n\to\infty}... uad \hbox{for all}\,\, n.$ - Let $\{f_n\}$ be a sequence of functions defined on $[a,b]$. If $f_n \to f$ u... on} **Questions from Chapter 03:** - Consider a sequence of functions $E_n:\mathbb{R}\to\mathbb{R}$ define
- Unit 04: Sequence and Series (Solutions) @math-11-kpk:sol
- ===== Unit 04: Sequence and Series (Solutions) ===== This is a forth unit of the book Mathematics 11 publis... is unit the students will be able to * Define a sequence (progression) and its terms. * Know that a sequence can be constructed from a formula or an inductive def... al and pascal sequences. * Define an arithmetic sequence. * Find the nth or general term of an arithmeti
- Question 2, Exercise 4.2 @math-11-nbf:sol:unit04
- lutions of Question 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mat... ==== Find the next three terms of each arithmetic sequence. $5,9,13, \ldots$ ** Solution. ** Give: $$5, 9,... 25 \end{align*} Thus, the next three terms of the sequence are $17$, $21$, $25$. =====Question 2(ii)===== Find the next three terms of each arithmetic sequence. $11,14,17, \ldots$ ** Solution. ** Given: $$11
- Metric Spaces (Notes) @notes
- erever //G// is open in //Y//. * Convergence of Sequence * Theorem: If $(x_n)$ is converges then limit ... $(x_n)$ is unique. * Theorem: (i) A convergent sequence is bounded. (ii) ii) If ${x_n}\to x$ and ${y_n}\to y$ then $d(x_n,y_n)\to d(x,y)$. * Cauchy Sequence * Theorem: A convergent sequence in a metric space (//X,d//) is Cauchy. * Subsequence * Theorem: (i)