Math 162

Class 24

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 \([a - r, a + r]\)

Adding Power Series

Given two power series, \(\displaystyle\sum_{n=0}^\infty c_n(x-a)^n\) which converges to \(f\), and \(\displaystyle\sum_{n=0}^\infty d_n(x-a)^n\) converging to \(g\), with radii of convergence \(R_1\) and \(R_2\), the series \[\displaystyle\sum_{n=0}^\infty (c_n + d_n)(x-a)^n\] has a radius of convergence at least the smaller of \(R_1\) and \(R_2\), and converges to \(f + g\).

Multiplying by \(b(x-a)^m\)

Let \(\displaystyle\sum_{n=0}^\infty c_n(x-a)^n\) be a power series converging on an interval \(I\) to the function \(f\).

Then the power series \(\displaystyle\sum_{n=0}^\infty bc_n(x-a)^{n+m}\) converges to \(b(x-a)^mf(x)\) on \(I\).

Substituting

Let \(\displaystyle\sum_{n=0}^\infty c_nx^n\) be a power series converging on an interval \(I\) to the function \(f\).

Then the power series \(\displaystyle\sum_{n=0}^\infty c_nb^nx^{mn}\) converges to \(f(bx^m)\) for any \(x\) such that \(bx^m \in I\).

Example

Find a power series for \(f(x) = \dfrac{1}{x^2 - 3x + 2}\).

Example

Find a power series for \(g(x) = \dfrac{x}{x^2 + 1}\)

Exercise

Find all derivatives of the polynomial

\[P(x) = 2x^5 - x^4 + 3x^3 + x^2 - 6x + 31\]

Derivative of Power Series

Let \(\displaystyle\sum_{n=0}^\infty c_n(x-a)^n\) be a power series converging to a function \(f\) with a radius of convergence \(R>0\).

Then the series \(\displaystyle\sum_{n=1}^\infty nc_n(x-a)^{n-1}\) converges to \(f'\) with the same radius of convergence \(R\).

Every power series centered at \(a\) with radius of convergence \(R>0\) is differentiable on \((a - R, a+R)\), and the derivative is again a power series centered at \(a\) with radius of convergence \(R\).

It is infinitely differentiable on \((a - R, a + R)\).

Example

We know that \(\displaystyle \frac{1}{1-x} = \sum_{n = 0}^\infty x^n\)

Exercise

Find the antiderivative of the polynomial

\[P(x) = 2x^5 - x^4 + 3x^3 + x^2 - 6x + 31\]

Integration of Power Series

Let \(\displaystyle\sum_{n=0}^\infty c_n(x-a)^n\) be a power series converging to a function \(f\) with a radius of convergence \(R>0\).

Then the antiderivative of \(f\) on \((a-R,a+R)\) is \[\int f(x)\;dx = C + \sum_{n=0}^\infty\frac{c_n}{n+1}(x-a)^{n+1}\]

With an index shift, that can be written as \[\int f(x)\;dx = C + \sum_{n=1}^\infty\frac{c_{n-1}}{n}(x-a)^{n}\]

Example

Start with the series with sum \(\displaystyle \frac{1}{1+x^2}\)