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In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence.
In uniform convergence, one is given \(ε > 0\) and must find a single \(N\) that works for that particular \(ε\) but also simultaneously (uniformly) for all \(x ∈ S\). Clearly uniform convergence implies pointwise convergence as an \(N\) which works uniformly for all \(x\), works for each individual \(x\) also.
A series sumf_n (x) converges uniformly on E if the sequence {S_n} of partial sums defined by sum_ (k=1)^nf_k (x)=S_n (x) (2) converges uniformly on E. To test for uniform convergence, use Abel's uniform convergence test or the Weierstrass M-test. If...
To determine uniform convergence, let u n(t) =tn and for suitably small r, let K n = sup t2[ 1+ r;1 ] jtj n = 1r=ˆ<1. Since P 1 n=1 K = 1 n=1 ˆ n is a geometric series with 0 ˆ<1, it converges. Hence by the Second Weierstrass Uniform Convergence Theorem (SWUCT), the convergence of the series P 1 n=0 t n is uniform on
If f(x) converges for all x ∈ A, we say that the sum given by eq. (1) is pointwise convergent over the interval x ∈ A. In this case, A is called the interval of convergence. A classic example is the infinite geometric series, ∞ 1. =.
Sequences and Series of Functions. unctions. There are many different ways to define the convergence of a sequence of functions, and different definitions lead to inequivalent types of co. vergence. We consider here two basic types: pointwise and uniform co. 5.1. Pointwise convergence.
The series \(\sum_{n=1}^\infty x^n\) converges uniformly on \([0,\frac{1}{2}]\). To prove this, use the uniform Cauchy Criterion. Let \(S_K(x)\) be the \(K\)-th partial sum, i.e., \(S_K(x) := x + x^2 + \cdots + x^K\). Then \(S_{K_1}(x) - S_{K_2}(x) = x^{K_2+1} + \cdots + x^{K_1}\) assuming that \(K_1 > K_2\). Then