Prove divergence of an alternating sequence
Prove that the divergence of the following sequence.$$s_n=\frac{(-1)^nn}{2n-1}$$The following is the sample answerNote that $\exists N\in\mathbb{N}\, s.t.\,\forall k\geq...
View Article$ f'(x) \leq f'\left(x + \frac{1}{n} \right) $ for all $ x \in \mathbb{R} $,...
Romanian Math Olympiad 1998Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that$f'(x) \leq f'\left(x + \frac{1}{n} \right)$for all $x \in \mathbb{R}$ and all positive...
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Minimizing CDF of a hypergeometric distribution at its expectation with...
Let $X_m \sim \mathcal {HG} (N,m,n)$, for which the expectation is given by $\mathbb E(X_m)=\frac{nm}{N}$. Assume that $N$ and $n$ are fixed, and we want to determine $m \in [N-1]$ that minimizes the...
View ArticleWhy are every structures I study based on Real number?
I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my...
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Implication of right-continuous step function on $\mathbb R$ not having limit...
Yeh in his book Real Analysis defines a right-continuous step function $f$ on a finite interval $[a, b) $ as of the form $f= \sum_{k=1}^n c_k\cdot \mathbf 1 _{I_k}, $ where $\sqcup_{k=1}^n I_k=[a, b) $...
View ArticleInterchangeable limits
The following exercise 2.2.9 is borrowed from Terence Tao's Analysis II, page 33. While I understand the problem completely, I lack the technique to attack it. So I will really appreciate a small hint...
View ArticleAbbott's Understanding Analysis Exercise 1.2.4.
Understanding Analysis Exercise 1.2.4.Produce an infinite collection of sets $A_1, A_2, A_3,\ldots$ with the property that every $A_i$ has an infinite number of elements, $A_i \cap A_j = \varnothing$...
View ArticleCan we partiton ℝ into sets $A$ and $B$, where the Lebesgue measures in every...
Suppose $\lambda$ is the Lebesgue measure on the Borel $\sigma$-algebra.Does there exist an example of sets $A,B\subset\mathbb{R}$, where:$A\cup B=\mathbb{R}$$A\cap B=\emptyset$for all non-empty open...
View ArticleMaclaurin series of an inverse function.
I have a problem I can't seem to really wrap around my head at the moment.Let $g(x)=2\tan(x)-x$, $|x|\le\cfrac{\pi}{2}$My task is to find the maclaurin series of the inverse function $g^{-1}(x)$ up...
View ArticleSolve the differential equation $f'=f/(f\circ f)$
I am completely stuck on the following differential equation:Find all differentiable functions $ f:\mathbb{R}_{>0}\to\mathbb{R}_{>0} $ such that:$$ f' = \frac{f}{f\circ f}. $$The identity...
View Article$n$th derivative of $\log(x)$
I am attempting to write down a closed from expression for$$\frac{d^n}{dx^n}\log(f(x))$$at $x=x_0$ with the additional assumptions that $f(x_0)=1$ and $f'(x_0)=0$. As mentioned in this post, the...
View ArticleHow to formally show these functions are solutions to these ODEs of finite...
How to formally show these functions are solutions to these ODEs of finite duration?Brief IntroI am trying to understand ODEs of finite duration, so I tried by tackling the easiest examples I found,...
View ArticleDoes the series $ \sum_{n=1}^{\infty}\left( 1-\cos\big(\frac{1}{n} \big)...
I'm having trouble determining whether the series:$$\sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right]$$converges.I have tried the root...
View Articlepractice papers for differentiation on R^n _ Real Analysis [closed]
I'm currently taking Real Analysis course, and we are taking differentiation of function from R^p to R^q, I searched for practice papers online, but i couldn't find any. Kindly, if any of you have...
View ArticleProve that sequence defined by $a_1 = \sqrt{2}, a_{n+1} = \sqrt{2 +...
Prove that the following sequence defined by the given recurrence relation is convergent:\begin{align*} a_1 &= \sqrt{2} \\ a_{n+1} &= \sqrt{2 + \sqrt{a_n}} \\\end{align*}First, I can prove that...
View ArticleInfinite Series $\sum\limits_{n=1}^\infty\frac{(H_n)^2}{n^3}$ [closed]
How to prove that$$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$$H_n$ denotes the harmonic numbers.
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Solving $x'=-\text{sgn}(x)\sqrt{|x|}$: Uniqueness of solutions of finite...
Show that $$x(t) = \frac{\text{sgn}(x(0))}{4}\left(2\sqrt{|x(0)|}-t\right)^2\cdot\theta\!\left(2\sqrt{|x(0)|}-t\right)$$ is a solution to$$x'=-\text{sgn}(x)\sqrt{|x|}.$$Is the solution unique?This...
View ArticleProve that $\|f\|_2 :=\left( \int_0^1 |f(t)|^2 dt \right)^{1/2}$ defines a norm
This question is an exercise from the first chapter of Topology of metric spaces by S.KumaresanLet $X := [0,1]$, the closed unit interval. Then$$\|f\|_2 := \left(\int_0^1 |f(t)|^2...
View ArticleDoes this argument work? (Cesaro Means)
Question I am trying to do: Prove that if $(x_n)$ is a convergent sequence, then the sequence given by the averages $$ y_n=\frac{x_1+x_2+\dots+x_n}{n} $$ also converges to the same limit.In the...
View ArticleA sequence of finite recursive sequences [closed]
For a positive integer $n$ define a sequence $x_0=\frac{1}{2}$ and for $1 \leq k \leq n$ define $$x_k=x_{k-1}+\frac{x_{k-1}^2}{n}$$ Show that $x_n$ lies in the open interval $\left(1-\frac{1}{n},...
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