### calculus - Convergence of Infinite root

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

### calculus - How do I determine the distance between v and PQ when v =[2,1,2] and PQ = [1,0,3]? P = [0,0,0] Q = [1,0,3]

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

### calculus - Using Taylor Polynomials Programmatically in Maple

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

### calculus - Proof of convergence of a telescoping series

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

### self learning - What are your thoughts on Thomas' Calculus?

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

### calculus - Antiderivative of $g(x)dg(x)$

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