Homework Statement
Prove that for any n \in \mathbb{N} and x \in \mathbb{R}, we have
\sum_{k = 0}^{n} {\cos{(kx)}} = \frac{1}{2}+ \frac{\cos{(nx)} - \cos{[(n+1)x]}}{2 - 2\cos {x}}
Homework Equations
None I can think of.
The Attempt at a Solution
Try induction. The result holds if n = 0...
Homework Statement
Let A = \{(x, y, z) \in \mathbb{R}^n : 0 \lt x \leq 1, 0 \lt y \leq 1 - x^2, 0 \lt z \leq x^2 + y\}. Define f : A \rightarrow \mathbb{R} by f(x, y, z) = y for each (x, y, z) \in A. Accept that Fubini's theorem is applicable here. Find \int_A f.
Homework Equations
Fubini's...
Does your school's math curriculum "satisfy" you?
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"1-norm" is larger than the Euclidean norm
Define, for each \vec{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n, the (usual) Euclidean norm \Vert{\vec{x}}\Vert = \sqrt{\sum_{j = 1}^n x_j^2} and the 1-norm \Vert{\vec{x}}\Vert_1 = {\sum_{j = 1}^n |x_j|}.
How can we show that, for all \vec{x} \in...
Call {a1, a2, a3, ...} = {an} a "convergent sequence" if
\exists L \in \mathbb{R} : \quad \forall \epsilon > 0 \quad \exists N \in \mathbb{N} : (\forall n > N \quad (n > N \implies |a_n - L| < \epsilon))
in which case we write \lim_{n \rightarrow \infty} a_n = \lim a_n = L. Of course this...
Splitting the reciprocal of a factorial into a sum of reciprocals of positive integer
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Homework Statement
We are required to sketch a (reasonably accurate) picture of a rational function f(x) = P(x)/Q(x) with P, Q polynomials in x and Q nonzero. We know that the roots of Q(x) are, say, x1, x2, etc. and so f(x) is (typically) asymptotic to the vertical lines x = xk for each k...
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I'm looking for an example of the squeeze theorem involving elementary real-valued...
To compute this, we’ll make use of Euler’s formula cis(x) = eix = cos(x) + i·sin(x):
ei(π/2) = cos(π/2) + i·sin(π/2) = i, and exponentiating by i we get
ii = (ei(π/2))i = ei·i(π/2) = e-π/2 ∈ ℝ.
But we also have cis(2πk + π/2) = i, k ∈ ℤ. Thus by the same logic, we get ii = e-(2πk + π/2)...
From the definition of an open set as a set containing at least one neighborhood of each of its points, and a closed set being a set containing all its limit points, how can we show that the complement of an open set is a closed set (and vice versa)? Usually this is taken as a definition, but...
I ran into some difficulties trying to show "prove" the contrapositive law (CPL). I remember in first year my professor showed that P ⇒ Q is logically equivalent to ¬Q ⇒ ¬P by showing that the truth tables for both statements were the same for all possible truth values of P and Q.
Statement...
Sometimes I find that while a proof can be carried out "by contradiction", this is a pretty sloppy way of proving the desired statement. I wonder if the "←" direction of the following proof is sound presentation of proof by contradiction.
Statement. An integer is even if and only if its square...