Uniform Convergence of Series of Functions - Revision history

Lila: /* Licensing */

2021-10-28T23:08:51Z

Licensing

Lila: /* The Weierstrass M-Test for Uniform Convergence of Series of Functions */

2021-10-28T23:07:00Z

The Weierstrass M-Test for Uniform Convergence of Series of Functions

Lila: /* The Weierstrass M-Test for Uniform Convergence of Series of Functions */

2021-10-27T20:15:35Z

The Weierstrass M-Test for Uniform Convergence of Series of Functions

Lila: /* The Weierstrass M-Test for Uniform Convergence of Series of Functions */

2021-10-27T20:14:32Z

The Weierstrass M-Test for Uniform Convergence of Series of Functions

Lila: /* The Weierstrass M-Test for Uniform Convergence of Series of Functions */

2021-10-27T20:13:59Z

The Weierstrass M-Test for Uniform Convergence of Series of Functions

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Licensing

Lila: /* Cauchy's Uniform Convergence Criterion for Series of Functions */

2021-10-27T19:54:51Z

Cauchy's Uniform Convergence Criterion for Series of Functions

Lila: /* Cauchy's Uniform Convergence Criterion for Series of Functions */

2021-10-27T19:54:16Z

Cauchy's Uniform Convergence Criterion for Series of Functions

Lila: /* Cauchy's Uniform Convergence Criterion for Series of Functions */

2021-10-27T19:53:09Z

Cauchy's Uniform Convergence Criterion for Series of Functions

Lila: /* Cauchy's Uniform Convergence Criterion for Series of Functions */

2021-10-27T19:51:50Z

Cauchy's Uniform Convergence Criterion for Series of Functions

← Older revision		Revision as of 23:08, 28 October 2021
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	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [http://mathonline.wikidot.com/pointwise-convergent-and-uniformly-convergent-series-of-func Pointwise Convergent and Uniformly Convergent Series of Functions, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/pointwise-convergent-and-uniformly-convergent-series-of-func Pointwise Convergent and Uniformly Convergent Series of Functions, mathonline.wikidot.com] under a CC BY-SA license
		+	* [https://en.wikipedia.org/wiki/Uniform_convergence Uniform convergence, Wikipedia] under a CC BY-SA license

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 <li>So the <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> converges for each <math>x \in X</math> by the comparison test. <math>\blacksquare</math></li>
 </ul>
 ==Licensing==
 Content obtained and/or adapted from:
 * [http://mathonline.wikidot.com/pointwise-convergent-and-uniformly-convergent-series-of-func Pointwise Convergent and Uniformly Convergent Series of Functions, mathonline.wikidot.com] under a CC BY-SA license

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 <p>Recall that if <math>(f_n(x))_{n=1}^{\infty}</math> is a sequence of real-valued functions with common domain <math>X</math>, then we say that the corresponding series of functions <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is uniformly convergent if the sequence of partial sums <math>(s_n(x))_{n=1}^{\infty}</math> is a uniformly convergent sequence.</p>
 <p>We will now look at a very nice and relatively simply test to determine uniform convergence of a series of real-valued functions called the Weierstrass M-test.</p>
 <td><strong>Theorem 1:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of real-valued functions with common domain <math>X</math>, and let <math>(M_n)_{n=1}^{\infty}</math> be a sequence of nonnegative real numbers such that <math>\left| f_n(x) \right| \leq M_n</math> for each <math>n \in \mathbb{N}</math> and for all <math>x \in X</math>. If <math>\displaystyle{\sum_{n=1}^{\infty} M_n}</math> converges then <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> uniformly converges on <math>X</math>.</td>
 <ul>
 <li><strong>Proof:</strong> Suppose that there exists a sequence of nonnegative real numbers <math>(M_n)_{n=1}^{\infty}</math> such that for all <math>n \in \mathbb{N}</math> and for all <math>x \in X</math> we have that:</li>

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 </ul>
-<div class="math-equation" id="equation-2">\begin{align} \quad  \left| \sum_{n=1}^{\infty} f_n(x)  \right| \leq \sum_{n=1}^{\infty} \left| f_n(x) \right| \leq \sum_{n=1}^{\infty} M_n = M \end{align}</div>
+<math>\begin{align} \quad  \left| \sum_{n=1}^{\infty} f_n(x)  \right| \leq \sum_{n=1}^{\infty} \left| f_n(x) \right| \leq \sum_{n=1}^{\infty} M_n = M \end{align}</math>
 <ul>
 <li>So the <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> converges for each <math>x \in X</math> by the comparison test. <math>\blacksquare</math></li>

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 </ul>
-<div class="math-equation" id="equation-1">\begin{align} \quad \left| f_n(x) \right| \leq M_n \end{align}</div>
+<math>\begin{align} \quad \left| f_n(x) \right| \leq M_n \end{align}</math>
 <ul>
 <li>Furthermore, suppose that <math>\displaystyle{\sum_{n=1}^{\infty} M_n}</math> converges to some <math>M \in \mathbb{R}</math>, <math>M \geq 0</math>. Then we have that for all <math>x \in X</math>:</li>
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 <li>So the <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> converges for each <math>x \in X</math> by the comparison test. <math>\blacksquare</math></li>
 </ul>
 ==Licensing==
 Content obtained and/or adapted from:
 * [http://mathonline.wikidot.com/pointwise-convergent-and-uniformly-convergent-series-of-func Pointwise Convergent and Uniformly Convergent Series of Functions, mathonline.wikidot.com] under a CC BY-SA license

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 <p>We will now look at a nice theorem known as Cauchy's uniform convergence criterion for series of functions.</p>
 <table class="wiki-content-table">
-<tr>
-<blockquote style="background: white; border: 1px solid black; padding: 1em;"> <td><strong>Theorem 1:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of real-valued functions with common domain <math>X</math>. Then <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is uniformly convergent on <math>X</math> if and only if for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that <math>\displaystyle{ \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| < \varepsilon}</math> for all <math>p \in \mathbb{N}</math>.</td>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;"> <strong>Theorem 1:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of real-valued functions with common domain <math>X</math>. Then <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is uniformly convergent on <math>X</math> if and only if for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that <math>\displaystyle{ \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| < \varepsilon}</math> for all <math>p \in \mathbb{N}</math>.
 </blockquote>
 </table>
 <ul>

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 <ul>
 <li><strong>Proof:</strong> <math>\Rightarrow</math> Suppose that <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is uniformly convergent to some limit function <math>f(x)</math> on <math>X</math>. Let <math>(s_n(x))_{n=1}^{\infty}</math> denote the sequence of partial sums for this series. Then we must have that <math>\displaystyle{\lim_{n \to \infty} s_n(x) = f(x)}</math> uniformly on <math>X</math>. So, for <math>\varepsilon_1 = \frac{\varepsilon}{2}</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that:</li>
 </ul>
 <math>\begin{align} \quad \left| s_n(x) - f(x) \right| < \varepsilon \end{align}</math>
 <ul>
 <li>For any <math>p \in \mathbb{N}</math> let <math>m = n + p</math>. Then <math>m \geq N</math> and so:</li>
 </ul>
 <math>\begin{align} \quad \quad  \left| \sum_{k=1}^{m} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| =  \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| =  \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| = \left| s_m(x) - s_n(x) \right| \leq \left| s_m(x) - f(x) \right| + \left| f(x) - s_n(x) \right| < \varepsilon_1 + \varepsilon_1 = \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon \end{align}</math>
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 <li>Let <math>m, n \geq N</math>. Assume without loss of generality that <math>m > n</math> and that <math>m = n + p</math> for some <math>p \in \mathbb{N}</math>. Then from above we see that for all <math>x \in X</math>:</li>
 </ul>
 <math>\begin{align} \quad \left| s_m(x) - s_n(x) \right| =  \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| = \left| \sum_{k=n+1}^{n+p} f_k(x) \right| < \varepsilon \end{align}</math>
 <ul>
 <li>So <math>(s_n(x))_{n=1}^{\infty}</math> converges uniformly by the Cauchy uniform convergence criterion for sequences of functions. So <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> converges uniformly on <math>X</math>. <math>\blacksquare</math></li>
 </ul>