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 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 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
