Articles of limits

$ S_n=\sum_{k=1}^n\frac{1}{k}$ then is $S_n$ bounded?

Let , $\displaystyle S_n=\sum_{k=1}^n\frac{1}{k}$. Which of the following is TRUE ? (A) $\displaystyle S_{2^n}\ge \frac{n}{2}$ for every $n\ge 1$. (B) $S_n$ is bounded sequence. (C) $\displaystyle|S_{2^n}-S_{2^{n-1}}|\to 0$ as $n\to \infty$. (D) $\displaystyle\frac{S_n}{n}\to 1$ as $n\to \infty$. As the series is divergent , so (B) is FALSE. If (C) is TRUE then $\{S_n\}$ is a Cauchy […]

“Multi-valued limit” $\lim_{(s,t)\to (\frac{1}{2}, \frac{1}{2})} \frac{t-s}{1-2s}$

I’m working on a larger problem and it would be very helpful if the following were true: For $0\leq s < \frac{1}{2}$ and $s \leq t \leq 1-s$, for every $\varepsilon > 0$ there exists a $\delta > 0$ such that if $\sqrt{(t-\frac{1}{2})^{2}+(s-\frac{1}{2})^{2}} < \delta$ then either $$\left|\frac{t-s}{1-2s} \right| < \varepsilon \text{$\quad$ or $\quad$} \left|\frac{t-s}{1-2s} […]

Formal proof for $\lim_{x\to\infty}x\exp(-x) =0$

I intuitively understand that $$\lim_{x\rightarrow\infty} xe^{-x}=0$$ as the $e^{-\infty}$ approaches zero faster than $x$ approaches infinity. But this requires one to have a knowledge of the property of exponents. Is there any way to prove this formally that is mathematically sound? For example, how would the person without the knowledge of the property of exponents […]

Limit of a sequence of averaged numbers?

Let $a_0 = 0$, $a_1 = 1$, and $a_n = \frac{a_{n-1}+a_{n-2}}{2}$ for all $n \ge 2$. Consider $\lim \limits_{n \to \infty} a_n$. Using a quick python script I found that for large $n$ $a_n$ tends to $\frac{2}{3}$. How do I prove this result?

Evaluation of $\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!}$

I have problems evaluating the following limit: $$\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!}$$ What causes problems in particular is that I am unsure how to behave when there is a sum which becomes a series. I am aware that I need to evaluate all limits at the same time. So, I am not […]

Limit of a complex number over its conjugate, as z approach the infinity.

this question must be pretty easy but i´m just taking my firs course in complex variables, i need to find the limit of Z over it conjugate as z reach the infinity. I need to know how to do it in a rigorous way and informal way, so i need your help. By formal, i […]

Evaluate the limit of $\sum\limits_{i=1}^n\frac{2}{n}\left$ when $n\to\infty$

How would I evaluate: $$\lim_{n \to \infty}\sum_{i=1}^n\frac{2}{n}\left[ \left( \frac{3i}{n} \right)^3+\frac{5i}{n}+2\right]$$ So far, I have cubed the inner bracket and taken the $\dfrac{27}{n^3}$ outside to get: $$\lim_{n \to \infty} \sum_{i=1}^n \frac{54}{n^4}\left(i^3+\frac{5n^2i}{27}+\frac{2n^3}{27}\right).$$ Then I would take the summation of each individual term while substituting the summation of $i^3$ from $i=1$ to $n$ as $(n^2(n+1)^2)/4$. What is confusing […]

Prove that $\lim_{n\to\infty} H_n/n = 0$ ($H_n$ is the $n$-th harmonic number) using certain techniques

I can’t seem to use certain methods such as $\varepsilon$-N, L’Hôspital’s Rule, Riemann Sums, Integral Test and Divergence Test Contrapositive or Euler’s Integral Representation to prove that $\lim_{n-> \infty} \frac{H_n}{n} = 0$ where $H_n$ is the nth Harmonic number $= \sum_{i=1}^{n} \frac{1}{i} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + … + \frac{1}{n}$? I […]

Does uniform continuity imply the interchangeability of the order of the limits?

I am learning Uniform Continuous in the Advanced Calculus class now. Today, teacher taught a very important theorem (he said) as following: Theorem: Let $A\subset M$ where M is a metric space, let $f_k:A\to N$ be a sequence of continuous functions, and suppose that $f_k \to f$ (uniformly on $A$). Then $f$ is continuous on […]

Absolute value and limit reasoning.

I am trying to develop my reasoning ability with absolute value. So, I wanted to know if the following reasoning is correct: Find $\lim_{x \to -6}\dfrac{2x+12}{|x+6|}$ By definition of absolute value we have $|x| = x$ when $x > 0$ and $|x| = -x$ when $x<0$ So for the above limit we can consider the […]