Math 162

Class 23

Sequences of Functions

  • Infinite sequence of functions \(f_i\), \(i = k, k+1, \dots\)
  • All defined on a common domain \(D\).
  • For each \(x \in D\) we get a sequence \(f_i(x)\), \(i = k, k+1, \dots\)
  • Is this sequence convergent?
  • If it is, what is the \(\displaystyle\lim_{i\to\infty} f_i(x)\)?
  • Let \(E\) be the set of all \(x\in D\) such that \(\displaystyle\lim_{i\to\infty} f_i(x)\) exists.

Limit of Sequence

  • Let \(E\) be the set of all \(x\in D\) such that \(\displaystyle\lim_{i\to\infty} f_i(x)\) exists.
  • Define a function \(f\) with domain \(E\): \[f(x) = \lim_{i\to\infty} f_i(x)\]
  • We say that the function \(f\) is the limit of the sequence \((f_i)_{i = k, k+1, \dots}\), or that the sequence \((f_i)\) converges to \(f\) on the set \(E\).

Example

x yf 1 ( x )= xf 2 ( x )= x 2f 3 ( x )= x 3f 4 ( x )= x 4f 5 ( x )= x 5f 6 ( x )= x 6f 7 ( x )= x 7f 8 ( x )= x 8f 9 ( x )= x 9f 10 ( x )= x 10f 11 ( x )= x 11f 12 ( x )= x 12f 13 ( x )= x 13f 14 ( x )= x 14f 15 ( x )= x 15f 16 ( x )= x 16f 17 ( x )= x 17f 18 ( x )= x 18f 19 ( x )= x 19f 20 ( x )= x 20 f ( x )

Example

\(f_i(x) = x^i\) for \(i = 1, 2, \dots\)

  • If \(-1 < x < 1\), \(\displaystyle\lim_{i\to\infty}f_i(x) = 0\).
  • If \(x = 1\), \(\displaystyle\lim_{i\to\infty}f_i(x) = 1\).
  • If \(x \le -1\) or \(1 < x\), \(\displaystyle\lim_{i\to\infty}f_i(x)\) does not exist.

\[f(x) = \begin{cases} 0 & \text{ if $-1 < x < 1$}\\1 & \text{ if $x = 1$}\end{cases}\]

The sequence \((f_i)\) converges to \(f\) on \((-1,1]\).

Another Example

x yf 1 ( x )= exp − x 2 f 2 ( x )= exp − 2 x 2 f 3 ( x )= exp − 3 x 2 f 4 ( x )= exp − 4 x 2 f 5 ( x )= exp − 5 x 2 f 6 ( x )= exp − 6 x 2 f 7 ( x )= exp − 7 x 2 f 8 ( x )= exp − 8 x 2 f 9 ( x )= exp − 9 x 2 f 10 ( x )= exp − 10 x 2 f 11 ( x )= exp − 11 x 2 f 12 ( x )= exp − 12 x 2 f 13 ( x )= exp − 13 x 2 f 14 ( x )= exp − 14 x 2 f 15 ( x )= exp − 15 x 2 f 16 ( x )= exp − 16 x 2 f 17 ( x )= exp − 17 x 2 f 18 ( x )= exp − 18 x 2 f 19 ( x )= exp − 19 x 2 f 20 ( x )= exp − 20 x 2 f ( x )

Another Example

\(f_i(x) = e^{-ix^2}\) for \(i = 1, 2, \dots\)

  • If \(x \neq 0\), \(\displaystyle\lim_{i\to\infty}f_i(x) = 0\).
  • If \(x = 0\), \(\displaystyle\lim_{i\to\infty}f_i(x) = 1\).

\[f(x) = \begin{cases} 0 & \text{ if $x \neq 0$}\\1 & \text{ if $x = 0$}\end{cases}\]

The sequence \((f_i)\) converges to \(f\) on \((-\infty, \infty)\).

What can we say about \(f\)?

Not much!

Does not have to be continuous.

Does not have to be differentiable.

Sometimes we can say more.

Uniform Convergence

Definition: We say that a sequence of functions \((f_i)\) converges to a function \(f\) uniformly on a set \(K\) if for each \(\varepsilon > 0\) there exists an integer \(N\) such that for each \(x \in K\), \[\left\lvert f_i(x) - f(x)\right\rvert < \varepsilon \text{ if } i \ge N\].

Note that the integer \(N\) depends only on the sequence \((f_i)\) and \(\varepsilon\), not on \(x\)!

Some Facts

  • If each \(f_i\) is continuous on \(K\) and the sequence \((f_i)\) converges to \(f\) uniformly on \(K\), then \(f\) is continuous.
  • If each \(f_i\) is differentiable on \(K\) with derivative \(f'_i\), the sequence \((f_i)\) converges to a function \(f\) on \(K\), and the sequence \((f'_i)\) converges to a function \(g\) uniformly on \(K\), then \(f\) is differentiable on \(K\), and \(f'(x) = g(x)\) for every \(x \in K\).
  • If each \(f_i\) is Riemann integrable on an interval \(K=[a,b]\) and the sequence \((f_i)\) converges to \(f\) uniformly on \(K\), then \(f\) is Riemann integrable on \(K\), and \[\int_a^b f(x)\;dx = \lim_{i\to\infty}\int_a^b f_i(x)\;dx\]

Infinite Series

Given an infinite sequence of functions \(f_i\), \(i = k, k+1, \dots\), we define an infinite series of the functions \(f_i\) to be the sum \[\sum_{i=k}^\infty f_i(x).\]

The sequence of partial sums \((s_n)\), \(n = k, k+1, \dots\) is the sequence of functions defined as \[s_n(x) = \sum_{i = k}^n f_i(x).\]

If the sequence \((s_n)\) converges to a function \(s\) on a set \(E\), we say that the function \(s\) is the sum of the series, and we say that the series is convergent on the set \(E\).

\[\sum_{i=k}^\infty f_i(x) = s(x)\]

Example

  • \(f_i(x) = x^i\) for \(i = 0, 1, \dots\)
  • The series is \[\sum_{i=0}^\infty x^i\]
  • This is a geometric series, convergent if \(\left\lvert x\right\rvert < 1\).
  • The sum is \[\frac{1}{1-x}\]
  • \[\sum_{i=0}^\infty x^i = \frac{1}{1-x} \text{ for } -1 < x < 1\]

Example

x ys 0 ( x )= 1 s 1 ( x )= 1 + xs 2 ( x )= 1 + x + x 2s 3 ( x )= 1 + x + ·· · + x 3s 4 ( x )= 1 + x + ·· · + x 4s 5 ( x )= 1 + x + ·· · + x 5s 6 ( x )= 1 + x + ·· · + x 6s 7 ( x )= 1 + x + ·· · + x 7s 8 ( x )= 1 + x + ·· · + x 8s 9 ( x )= 1 + x + ·· · + x 9s 10 ( x )= 1 + x + ·· · + x 10s 11 ( x )= 1 + x + ·· · + x 11s 12 ( x )= 1 + x + ·· · + x 12s 13 ( x )= 1 + x + ·· · + x 13s 14 ( x )= 1 + x + ·· · + x 14s 15 ( x )= 1 + x + ·· · + x 15s 16 ( x )= 1 + x + ·· · + x 16s 17 ( x )= 1 + x + ·· · + x 17s 18 ( x )= 1 + x + ·· · + x 18s 19 ( x )= 1 + x + ·· · + x 19 s ( x )= 1 + x + ·· · = 1 1 − x

Power Series

Let \(c_n\) for \(n = 0, 1, 2, \dots\) be a sequence of numbers, and \(a\) a number. The series of functions \[\sum_{i=0}^\infty c_n(x-a)^n = c_0 + c_1(x - a) + c_2(x-a)^2 + c_3(x-a)^3 + \cdots\] is called a power series centered at \(a\).

When \(a = 0\) we just get \[\sum_{i=0}^\infty c_nx^n = c_0 + c_1x + c_2x^2 + c_3x^3 + \cdots\]

Example

\(\displaystyle\sum_{n=0}^\infty n!(x-3)^n\)

Example

\(\displaystyle\sum_{n=0}^\infty n(x-3)^n\)

Example

\(\displaystyle\sum_{n=1}^\infty (-1)^n\frac{x^n}{n}\)

Example

\(\displaystyle\sum_{n=0}^\infty \frac{(x+5)^n}{2^n}\)

Convergence of Power Series

Given the power series \[\sum_{i=0}^\infty c_n(x-a)^n = c_0 + c_1(x - a) + c_2(x-a)^2 + c_3(x-a)^3 + \cdots\] exactly one of the following will happen:

  1. The series converges at \(a\) only.
  2. The series converges on \((-\infty, \infty)\)
  3. There is a positive number \(R\) such that the series converges on an interval with end-points \(a - R\) and \(a + R\), and diverges outside of that interval.

The number \(R\) is called the radius of convergence of the series.

Convergence of Power Series

Let \[\sum_{i=0}^\infty c_n(x-a)^n = c_0 + c_1(x - a) + c_2(x-a)^2 + c_3(x-a)^3 + \cdots\] be a power series with a positive (possibly infinite) radius of convergence \(R\). Then

  • The series converges absolutely on \((a - R, a+R)\).
  • For any positive \(r < R\), the series converges uniformly on \([a - r, a + r]\)