Articles of calculus

How to find this limit with following constraints?

Question: If a,b are 2 positive , co-prime integers such that $$\lim _{n \rightarrow \infty}(\frac{^{3n}C_n}{^{2n}C_n})^\frac{1}{n}=\frac{a}{b}$$Then a+b?? I tried to break down the limit to: $$\frac{[(3n)(3n-1)\dot{}\dot{}\dot{})^\frac{1}{n}][n(n-1)(n-2)\dot{}\dot{}\dot{})^\frac{1}{n}]}{[2n(2n-1)(2n-2)\dot{}\dot{}\dot{}]^\frac{2}{n}}$$ but I’m lost ahead of it. Answer given in my text books is 43. Please guide me

Prove the function $f(x)=1-x-(1-\frac{a}{n}x)^n$ has a real solution between $(0,1]$

Consider the function $f(x)=1-x-(1-\frac{a}{n}x)^n$ where $a$ and $n$ are both constants such that $\frac{a}{n}\in[0,1]$ ( it represents some probability indeed). One obvious solution is $x=0$. I want to prove that such a function has a real solution between $(0,1]$ and this solution has the form that dependening on $a$ and independent of $n$. The only […]

Two simple series

I dont know how to calculate these two series: $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( 1-\cos \frac{\pi }{n} \right)} \\ \end{align}$$

Finding area of a triangle with integrals

I am suppose to find the area of a triangle using integrals with vertices 0,0 1,2 and 3,1 This gives me $y= 2x$ $y=\frac{1}{3}x$ $y= \frac{-1}{2}x+\frac{5}{2}$ for my slopes I know that I can calculate the area of the first part by finding $$\int_{0}^{1}2x-\frac{1}{3}x$$ and the second part by $$\int_{1}^{3}\frac{-1}{2}x+\frac{5}{2}-\frac{1}{3}x$$ The anti derivative of the […]

An inequality involving supremum and integral

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ does not depend on $r>0$.

Determining the best possible substitution for an integrand

What substitution is best used to calculate $$\int \frac{1}{1 + \sqrt{x^2 -1}}dx$$

Spivak's Calculus – Chapter 1 Question 1.3

I’m reading through Spivak’s Calculus, and am not sure where to start in proving the following: if $x^2 = y^2$, then $x = y$ or $x = -y$ Particularly given that only the following properties can be used to justify each step of the proof: Any hints on how to start would be much appreciated!

The limit of hazard rate $h(x)=A/(1-B)$ as $x$ approaches $\pm \infty$

Can we tell what happens to the limit as $x$ approaches $\pm \infty$ of a hazard rate $h(x)$ defined for unspecified or generalized density as: $$ h(x)=A/(1-B) $$ where $A=f(x)$ is the density function, and $B=F(x)$ is the CDF.

How can I find the following limit: $\lim\limits_{x\to \:4}\frac{\sqrt{x+5}-3}{x-4}$

Find the following limit:$$\lim\limits_{x\to \:4}\frac{\sqrt{x+5}-3}{x-4}$$ I tried to multiply by the conjugate and it did not work.

Spivak Calculus on Manifolds, Theorem 5-2

In the proof Theorem 5-2 of Spivak Calculus on Mannifolds how is \begin{align*} V_2\cap M=\{f(a):(a,0)\in V_1\}? \end{align*} (That $\{f(a):(a,0)\in V_1\}=\{g(a,0):(a,0)\in V_1\}$ is clear.) Edit: Due to a comment, Theorem 5-2 proves the equivalence of the following two definitions of a $k$-dimensional regular submanifold in $\mathbb{R}^n$. Definition 1. A subset $M\subset\mathbb{R}^n$ is a $k$-dimensional submanifold if […]