Recent content by Tetrinity

Ahh, figures it'd be something as simple as that. \sum P_{n}(-x)(-r)^{n} = (1 + 2rx + r^{2})^{\frac{1}{2}} = \sum P_{n}(x)r^{n} \Rightarrow \sum P_{n}(x)r^{n} = \sum P_{n}(-x)(-1)^{n}r^{n} and then by applying the identity theorem for power series we get P_{n}(x) = (-1)^{n}P_{n}(x) as...

Homework Statement Let P_{n}(x) denote the Legendre polynomial of degree n, n = 0, 1, 2, ... . Using the formula for the generating function for the sequence of Legendre polynomials, show that: P_{n}(-x) = (-1)^{n}P_{n}(x) for any x \in [-1, 1], n = 0, 1, 2, ... . Homework Equations...

The axiom was stated, albeit rather informally ("there is no largest natural number"), in my Analysis I class from the previous semester (this homework is for Analysis II, which has Analysis I as a pre-requisite). I think I'll go with it. Thanks for all of your help!

The only definition of closure I have is the one stated in the original post - that the closure of a set X is the set of all of its limit points. Looking further into this, there seem to be a LOT of definitions of "closure" in mathematics... I think I may need to stick to the definition we've...

I am not working with Dedekind cuts. I must admit my understanding of them is somewhat limited, but the topic has not arisen in my course thus far, so I would assume they are not required. I can find such a sequence by setting qn = a1a2a3...ak-1.akak+1...an where ai is the ith digit of x. Then...

Homework Statement As the title states, the problem asks to prove that the closure of the set of rational numbers is equal to the set of real numbers. The problem includes the standard definition of the rationals as {p/q | q ≠ 0, p,q ∈ Z} and also states that the closure of a set X ⊂ R is...