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...") |
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<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> | ||
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<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>. | ||
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<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> | ||
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<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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} and for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \in \mathbb{N}} such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq N} then:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \left| f_n(x) - f(x) \right| < \varepsilon \end{align}}
Also recall that a sequence of functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} with common domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is said to be uniformly convergent if for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \in \mathbb{N}} such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq N} then for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} we have that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \left| f_n(x) - f(x) \right| < \varepsilon \end{align}}
We will now extend the concept of pointwise convergence and uniform convergence to series of functions.
Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} be a sequence of functions with common domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . The corresponding series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} is said to be Pointwise Convergent to the sum function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} if the corresponding sequence of partial sums Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s_n(x))_{n=1}^{\infty}} (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{s_n(x) = \sum_{k=1}^n f_n(x) = f_1(x) + f_2(x) + ... + f_n(x)}} ) is pointwise convergent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} .
For example, consider the following sequence of functions defined on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1, 1)} :
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (f_n(x))_{n=1}^{\infty} = (x^{n-1})_{n=1}^{\infty} = (1, x, x^2, x^3, ...) \end{align}}
We now that this series converges pointwise for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in (0, 1)} since the result series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} x^{n-1}} is simply a geometric series to the sum function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{f(x) = \frac{1}{1 - x}}} .
Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} be a sequence of functions with common domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . The corresponding series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} is said to be Uniformly Convergent to the sum function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} if the corresponding sequence of partial sums is uniformly convergent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} .
The geometric series given above actually converges uniformly on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1, 1)} , 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
