Articles of limits

A limit about $a_1=1,a_{n+1}=a_n+$

Let the sequence $\{a_n\}$ satisfy $$a_1=1,a_{n+1}=a_n+[\sqrt{a_n}]\quad(n\geq1),$$ where $[x]$ is the integer part of $x$. Find the limit $$\lim\limits_{n\to\infty}\frac{a_n}{n^2}$$. Add: By the Stolz formula, we have \begin{align*} &\mathop {\lim }\limits_{n \to \infty } \frac{{{a_n}}}{{{n^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{a_{n + 1}} – {a_n}}}{{2n + 1}} = \mathop {\lim }\limits_{n \to \infty } \frac{{\left[ […]

Limit of product with unbounded sequence $\lim_{n\to\infty} \sqrt{n}(\sqrt{n}-1)=0$

This question already has an answer here: How to prove $\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n} – 1) = 0$? 5 answers

Limit of the sequence $\frac{1^k+2^k+…+n^k}{n^{k+1}}$

This question already has an answer here: Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$ 4 answers

Find the limit of $(\sqrt2-\root3\of2 )(\sqrt2 -\root4\of2 )…(\sqrt 2-\root{n}\of2 )$ without using the squeezing principle

$$\mathop {\lim }\limits_{n \to \infty } (\sqrt 2 – \root 3 \of 2 )(\sqrt 2 – \root 4 \of 2 )…(\sqrt 2 – \root n \of 2 )$$ I was able easily to find the limit of this series using the squeezing principle, but how can you find it without it? I’m guessing it’s an […]

Proving that $\left(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}\right)$ has a limit

$$x_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}$$ How can we prove that the sequence $(x_n)$ has a limit? I have to use the fact that an increasing sequence has a limit iff it is bounded from above. No more “advanced” tools can be used. It’s obvious that this sequence is increasing, but I am having trouble finding a bound.

Problem 7 IMC 2015 – Integral and Limit

I’m trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ And here is my approach: Since $A^\frac{1}{x}=\exp\left(\frac{\ln A}{x}\right)$, then using Taylor series for exponential function, we have $$\exp\left(\frac{\ln A}{x}\right)=1+\frac{\ln A}{x}+\frac{\ln^2 A}{x^2}+\frac{\ln^3 A}{x^3}+\cdots=1+\frac{\ln A}{x}+\sum_{k=2}^\infty\frac{\ln^k A}{x^k}$$ Hence, integrating term by term is trivial. […]

Continuity of $\frac{2xy}{x^2+y^2}$ at $(0,0)$

Given a Heaviside function $$f(x,y)=\begin{cases}\frac{2xy}{x^2+y^2}, &x^2+y^2 \neq 0\\0 ,&x^2+y^2=0 \end{cases}$$ Letting $a$ and $b$ be fixed constants, show that for all values of $a$ and $b$, including $0$, the one variable functions $g(x)=f(x,b)$ and $h(y)=f(a,y)$ are both continuous on the entire real line. And how to determine whether the function is continuous at $(0,0)$. What […]

Does this condition on the sum of a function and its integral imply that the function goes to 0?

Consider a bounded real-valued function $S:\mathbf{R}\to\mathbf{R}$ so that $$\lim_{x\to\infty} \left( S(x) + \int_1^x \frac{S(t)}{t}dt\right)$$ exists and is finite. Can one say that $\lim_{x\to\infty} S(x)=0$?

Limit behavior of a definite integral that depends on a parameter.

Let $A>0$ and $0\le \mu \le 2$. Consider a following integral. \begin{equation} {\mathcal I}(A,\mu) := \int\limits_0^\infty e^{-(k A)^\mu} \cdot \frac{\cos(k)-1}{k} dk \end{equation} By substituting for $k A$ and then by expanding the cosine in a Taylor series about zero I have shown that: \begin{equation} {\mathcal I}(A,\mu) = \frac{1}{\mu} \sum\limits_{n=1}^\infty \frac{(1/A)^n}{n!} \cos\left(\frac{\pi}{2} n\right) \cdot \Gamma\left(\frac{n}{\mu}\right) \end{equation} […]

Find $\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}}$ using Riemann sums

Find $$\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}}$$ using Riemann sums. I got $$\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}} = \lim_{n \to \infty} \sum^n_{k=1} {1 \over {n^2}} {1 \over {(1+({k \over n})^2)^2}} $$ Now, this is not the classic $\lim_{n \to \infty} \sum^n_{k=1} {1 \over {n}} {1 \over {1+({k \over n})^2}}$ that […]