### calculus - Convergence of Infinite root

How can I determine if the following series converges?enter image description here...Read more

How can I determine if the following series converges?enter image description here...Read more

What I have tried already: d = |v||PQ|sin("Theta") Now, I need to determine what theta is, so I set up a position on a makeshift graph, the graph I made was on the xy plane only as the z plane complicates things needlessly for finding theta. So, I ended up with an acute angle, and if the angle is acute, then I have to find theta which according to dot product facts is greater than 0. I do not have access to theta, so I used the same princples from cross dots. u * v = |u||v|cos("theta") but in this case, u and v are PQ and v. A vector is a vecto...Read more

I am trying to use a Taylor polynomial programmatically in Maple, but the following does not seem to work...T[6]:=taylor(sin(x),x=Pi/4,6);convert(T[6], polynom, x);f:=proc(x) convert(T[6], polynom, x);end proc;f(1);All of the following also do not work:f:=convert(T[6], polynom);f:=convert(T[6], polynom, x);f:=x->convert(T[6], polynom);f:=x->convert(T[6], polynom, x);.Is there a way of doing this without copying and pasting the output of convert into the definition of f?...Read more

Show that the telescoping series below converges if and only if the $\lim_{j\to\infty} c_j$ is defined and finite. $$\sum_{j=1}^{\infty} c_j - c_{j+1}$$Not really sure where to start for this, proofs are nowhere near my strong suit. Would $c_j$ not be a constant? Why would I take the limit of a constant?Gotta go to class now, will check back this afternoon. Thanks in advance.Edit: my progress (from a reply below) that anyone can comment on:I'm currently trying to look at this and still not seeing it. Here is what I've played with. As you wo...Read more

I come from a country where international English books aren't easily available, and books published in my language are not at all useful. I wanted to start self-studying calculus and other higher mathematics. I have never touched calculus before and I really want to 'master' the subject (as in gain as much understanding as possible).I bought Thomas' Calculus, because that's the only one I could find. Tried finding Spivak (heard it's good), but no luck.I want to know : what's the best way of studying calculus, and how should I approach Thomas' ...Read more

I am reading a book by Shreve "Stochastic Calculus for Finance II" and after computing a stochastic integral $\int_{0}^{T}W(t)dW(t)$ where $W(t)$ is a Brownian motion he compares it to the integral$$\int_{0}^{T}g(t)dg(t) = \int_{0}^{T}g(t)g^\prime (t)dt = 0.5g^2(T),$$where $g(t)$ is a differentiable function with $g(0)=0$. I don't get the fact that $\int g(t)g^\prime (t)dt = 0.5g^2(t)$. For me the right hand side is equal to $\int g(t)dt$, without the $g^\prime (t)$ term. Thanks in advance....Read more

$$f(\phi) = \frac{{ n_1\choose x }{ n_2 \choose t-x } \phi^x}{\sum_{u=1}^t { n_1\choose u }{ n_2 \choose t-u } \phi^u}$$I'm trying to maximize the above function, but I am having trouble.I start by taking the log of the function and taking the derivative. Doing so gives me$$ \frac{x}{\phi} - \frac{\sum_{u=1}^t { n_1\choose u }{ n_2 \choose t-u } u \phi^{u-1}}{\sum_{u=1}^t { n_1\choose u }{ n_2 \choose t-u } \phi^u}$$Which after setting equal to 0 gives$$\frac{\sum_{u=1}^t { n_1\choose u }{ n_2 \choose t-u } u \phi^{u}}{\sum_{u=1}^t { n_1\choose...Read more

$$\int_0^\infty \frac{x^{\alpha}\ln x}{x^2+1}\,dx=\frac{\pi^2}{4} \frac{\sin(\pi \alpha/2)}{\cos^2(\pi \alpha/2)}$$ where $0 < \alpha < 1$.Answer: When i put this term in my integral calculator, it gave me very lengthy answer involving polylogarithm functions....Read more

My question is about understanding a step of the answer of the following question. show that $\limsup_{n\rightarrow\infty} \frac{\Sigma_{k=1}^{n} X_k}{n}<\infty$ a.s.I want to know how the following step has derived in the answer of the above question $\int_0^{\infty} P[Y>t]dt = \sum_{n=0}^{\infty} \int_{mn}^{m(n+1)} P[Y>t]dt $Is there any calculus theorem /definition behind this ? I am very keen to learn mathematical statistics and the understanding of this step will really helps me to solve lot of similar problems.I cannot directl...Read more

Given $C \subset \mathbb{R}^n$ a convex set, $x \in \overline{C}$ and $y \in \mbox{int} \;C$. Show that $(x, y] \subset \mbox{int}\; C$.I know the definition about convex set and I know that I need to prove for all point $p \in (x,y]$, there is a $\delta > 0$ such that $ p \in B(p,\delta) \subset C$.But I couldn't to prove this. I was thinking this a long time because this is apparently easy....Read more

I have the following static game with complete information. First of all the players play the static game depicted above infinitely repeated. They discount payoffs with the common discount factor $\delta$. And I am interesting in supporting (T,R),(T,R),...) as subgame perfect equilibrium I want to calculate the minimal discount factor needed so that my strategy supports this outcome. And secondly, this static game is assumed to be finite.y related. Now, I am I tested in supporting ((T,L),(D,R),...,(T,L), (D,R)) as a subgame perfect equilibrium....Read more

I have a set $$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$where $y>0$ is any number. And I want to show that this set B is closed bounded and convex.————-My Solution is as follows:First of all, I show this set in figure like that (but I am not sure about correctness, please tell me if it is wrong) Figure of the set is here. Is that figure true? Secondly I need to show that the set $B$ is closed.My idea: we know that any set in $R^2_+$ is closed. So, I try to show that the Set $B$ is closed. I guess I need to show that the compleme...Read more

I need to find the sum of the following series $\sum\limits_{n=1}^\infty \frac{X_n}{e^{n}}$ , where $X_n$ be I.I.D ,$P(X_n=1)=\frac12$ or $P(X_n=-1)=\frac12$Using the ratio test, it can be shown that this series is convergent.Because,$\lim \left| \frac{a_{n+1}}{a_n} \right|= lim \left| \frac{{1}}{e} \right|$<1 .But is there any method to find the actual sum ?? can i split this sum as $\sum\limits_{n=1}^\infty \frac{1}{e^{n}}$ - $\sum\limits_{n=1}^\infty \frac{1}{e^{n}}$ since $X_n$ can only take 1 and -1 ?...Read more

Are there any hard and fast rules for finding the sum of an infinite series? What suggestions would you have for someone new to learning them? Memorize identities? Also, what are the best resources for learning about them? I have searched for resources on Google, but they all give different examples with different ways to arrive at answers which have further confused me....Read more

Please don't give me the answer - I only want a hint.$$\lim_{x \to \pi}\frac{e^{\sin x} - 1}{x - \pi}$$This is a "Problems Plus" question from Stewart's Early Transcendentals (specifically, chapter 3 question 15).I have no idea how to solve this limit - I didn't even think it existed at first because the limit of the numerator is $-1$ and the limit of the denominator is $0$, but Wolfram Alpha goes against this logic in saying that the limit is $-1$.I've tried transforming this limit into the definition of a derivative for an easier time but hav...Read more