Difference between revisions of "Uniform Convergence of Series of Functions"
(Created page with "==Pointwise Convergent and Uniformly Convergent Series of Functions== <p>Recall that a sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> with common domain <math>X</m...") |
|||
| Line 1: | Line 1: | ||
| − | |||
<p>Recall that a sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> with common domain <math>X</math> is said to be pointwise convergent if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then:</p> | <p>Recall that a sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> with common domain <math>X</math> is said to be pointwise convergent if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then:</p> | ||
| Line 8: | Line 7: | ||
<p>We will now extend the concept of pointwise convergence and uniform convergence to series of functions.</p> | <p>We will now extend the concept of pointwise convergence and uniform convergence to series of functions.</p> | ||
| + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | ||
<strong>Definition:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. The corresponding series <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is said to be <strong>Pointwise Convergent</strong> to the sum function <math>f(x)</math> if the corresponding sequence of partial sums <math>(s_n(x))_{n=1}^{\infty}</math> (where <math>\displaystyle{s_n(x) = \sum_{k=1}^n f_n(x) = f_1(x) + f_2(x) + ... + f_n(x)}</math>) is pointwise convergent to <math>f(x)</math>. | <strong>Definition:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. The corresponding series <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is said to be <strong>Pointwise Convergent</strong> to the sum function <math>f(x)</math> if the corresponding sequence of partial sums <math>(s_n(x))_{n=1}^{\infty}</math> (where <math>\displaystyle{s_n(x) = \sum_{k=1}^n f_n(x) = f_1(x) + f_2(x) + ... + f_n(x)}</math>) is pointwise convergent to <math>f(x)</math>. | ||
| + | </blockquote> | ||
<p>For example, consider the following sequence of functions defined on the interval <math>(-1, 1)</math>:</p> | <p>For example, consider the following sequence of functions defined on the interval <math>(-1, 1)</math>:</p> | ||
| Line 15: | Line 16: | ||
<p>We now that this series converges pointwise for all <math>x \in (0, 1)</math> since the result series <math>\sum_{n=1}^{\infty} x^{n-1}</math> is simply a geometric series to the sum function <math>\displaystyle{f(x) = \frac{1}{1 - x}}</math>.</p> | <p>We now that this series converges pointwise for all <math>x \in (0, 1)</math> since the result series <math>\sum_{n=1}^{\infty} x^{n-1}</math> is simply a geometric series to the sum function <math>\displaystyle{f(x) = \frac{1}{1 - x}}</math>.</p> | ||
| + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | ||
<strong>Definition:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. The corresponding series <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is said to be <strong>Uniformly Convergent</strong> to the sum function <math>f(x)</math> if the corresponding sequence of partial sums is uniformly convergent to <math>f(x)</math>. | <strong>Definition:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. The corresponding series <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is said to be <strong>Uniformly Convergent</strong> to the sum function <math>f(x)</math> if the corresponding sequence of partial sums is uniformly convergent to <math>f(x)</math>. | ||
| + | </blockquote> | ||
<p>The geometric series given above actually converges uniformly on <math>(-1, 1)</math>, though, showing this with the current definition of uniform convergence of series of functions is laborious. We will soon develop methods to determine whether a series of functions converges uniformly or not without having to brute-force apply the definition for uniform convergence for the sequence of partial sums.</p> | <p>The geometric series given above actually converges uniformly on <math>(-1, 1)</math>, though, showing this with the current definition of uniform convergence of series of functions is laborious. We will soon develop methods to determine whether a series of functions converges uniformly or not without having to brute-force apply the definition for uniform convergence for the sequence of partial sums.</p> | ||
Revision as of 12:34, 27 October 2021
Recall that a sequence of functions with common domain is said to be pointwise convergent if for all and for all there exists an such that if then:
Also recall that a sequence of functions with common domain is said to be uniformly convergent if for all there exists an such that if then for all we have that:
We will now extend the concept of pointwise convergence and uniform convergence to series of functions.
Definition: Let be a sequence of functions with common domain . The corresponding series is said to be Pointwise Convergent to the sum function if the corresponding sequence of partial sums (where ) is pointwise convergent to .
For example, consider the following sequence of functions defined on the interval :
We now that this series converges pointwise for all since the result series is simply a geometric series to the sum function .
Definition: Let be a sequence of functions with common domain . The corresponding series is said to be Uniformly Convergent to the sum function if the corresponding sequence of partial sums is uniformly convergent to .
The geometric series given above actually converges uniformly on , though, showing this with the current definition of uniform convergence of series of functions is laborious. We will soon develop methods to determine whether a series of functions converges uniformly or not without having to brute-force apply the definition for uniform convergence for the sequence of partial sums.
Licensing
Content obtained and/or adapted from:
- Pointwise Convergent and Uniformly Convergent Series of Functions, mathonline.wikidot.com under a CC BY-SA license
