lacunary - Mathnotes

Taylor and Maclaurin Series

Definition: Power Series \@{power-series}

A power series in powers of z−z0 is a series of the form

∑n=0∞an(z−z0)n=a0+a1(z−z0)+a2(z−z0)2+⋯,

where z is a complex variable, a0,a1,…, are complex constants, called the coefficients of the series, and z0 is a complex constant, called the center of the series.

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Theorem: Taylor Expansion Theorem \@{taylor-expansion-theorem}

Let f be analytic in a domain D and z0 be a point in D. Then f can be expanded in a power series

f(z)=f(z0)+f′(z0)(z−z0)+f″(z0)2!(z−z0)2+⋯,

valid in all circles ‖z−z0‖<r containing only points of D.

Definition: Taylor series \@{taylor-series}

The expansion

f(z)=∑n=0∞f(n)(z0)n!(z−z0)n

is called the Taylor series of f about z0.

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Definition: Maclaurin series \@{maclaurin-series}

The special case of the Taylor series in which z0 = 0

f(z)=∑n=0∞f(n)(0)n!zn

is called the Maclaurin series of f.

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Definition: Circle of Convergence \@{circle-of-convergence}

The circle ‖z−z0‖< in which the Taylor series converges to the function is called the circle of convergence for the Taylor series.

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Theorem \@{theorem-7}

Every power series representation of (or, Taylor series for) an entire-function has an infinite radius of convergence.

Theorem: A theorem about the uniqueness of Taylor series as @power-series expansions \@{theorem-8}

If f has a power series expansion about a point z0 with nonzero radius of convergence, it must be the Taylor series about z0.

Theorem: A theorem about radius of convergence \@{theorem-9}

The radius of convergence of the Taylor series for a function f(z) about a point z0 is the distance from z0 to the nearest singularity of f(z).

Here are some important and useful Maclaurin series that we can often use to find those of other functions:

ez=1+z+z22!+z33!+⋯=∑n=0∞1n!zn,|z|<∞,

sin⁡z=z−z33!+z55!−z77!+⋯=∑n=0∞(−1)n(2n+1)!z2n+1,|z|<∞,

cos⁡z=z−z22!+z44!−z66!+⋯=∑n=0∞(−1)n(2n)!z2n,|z|<∞,

11−z=1+z+z2+z3+⋯=∑n=0∞zn,|z|<1.

Finding the Center and Radius of Convergence

Given a power series, we often want to know the center and radius of its circle of convergence. Finding its center is easy, from the form

∑n=0∞an(z−z0)n,

the center is simply z0. When z=z0, all terms in the series become 0 and so the power series always converges there.

To find the radius of convergence, we can shift the power series to be centered at 0, which makes it easier to manipulate.

To do this, let w=z−z0 (or, if we have something like ∑n=0∞an(z−z0)2n, then let w=(z−z0)2.

Now, the series becomes

∑n=0∞anwn.

Now, we can proceed with either to the ratio test or the root test. First, the root test.

|an+1wan+1anwn|=|an+1an||w|.

Here we can see that w contributes only a factor of |w|, which doesn't depend on n. So, when we take the limit as n→∞, all of the n-dependence is in the coefficient ratio |an+1/an|. Thus,

L=limn→∞|an+1an|⟹series converges if|w|<1/L.

So, our radius of convergence for our w series here is 1/L. We have to bring z back, so for example, if we had something like w=(z−z0)mn, then our radius of convergence would be (1/L)1/m. We get that by

|w|<R⟹|zm|<R⟹|z|<R1/m.

Similarly with the root test, we end up with something like

|w|<1lim sup|an|1/n.

Binomial Series

Theorem: Binomial Series \@{binomial-series}

For complex z and complex α, we have

(1+z)α=∑n=0∞(αn)zn.

If |z|<1, the series converges absolutely for any any complex α, including negative integers.

Other convergence conditions are listed here.

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Remark \@{remark-11}

This can be viewed as an extension of the Binomial Theorem.