6 - Power series

Question 1

What is the Taylor series?

Answer 1

A Taylor Series is an expansion of some function into an infinite sum of terms, where each term has a larger exponent like x, x2, x3, etc.

Taking the form:

f(x) = ∞∑(n=0) (f^n (a) / n!) * (x−a)^n

Question 2

What is the Taylor series expansion of exp (x)?

Answer 2

∞∑(n=0) x^n / n!

Question 3

What is the Taylor series expansion of cos (x)?

Answer 3

∞∑(n=0) ( (-1)^n / (2n)! ) * x^2n

Question 4

What is the Taylor series expansion of sin (x)?

Answer 4

∞∑(n=0) ( (-1)^n / (2n+1)! ) * x^(2n+1)

Question 5

What is the Taylor series expansion of 1/(1-x)?

Answer 5

∞∑(n=0) x^n

Question 6

What is a power series?

Answer 6

A power series about a, or just power series, is any series that can be written in the form,

∞∑(n=0) c_n * (x−a)^n

where a and c_n are real numbers. The c_n’s are often called the coefficients of the series. Power series have very nice convergence properties.

Question 7

What is the Cauchy-Hadamard Theorem?

Answer 7

The Cauchy-Hadamard Theorem concerns an expression for the radius of convergence. Every power series either converges for all values x or there exists a number R ϵ [0, ∞) such that it

• converges absolutely if |x| < R
• diverges if |x| > R

If it converges for every x we set R := ∞. Moreover,

R = 1 / [lim sup (n->∞) (|a_n|)^1/n]

Where by convention we set 1/∞ := 0 and 1/0 := ∞

Question 8

What does := mean?

Answer 8

It means set equal to, or is defined to be equal to

Question 9

What is the radius of convergence?

Answer 9

As discussed in the Cauchy-Hadamard theorem. The radius of convergence of a series is a number R that the power series will converge for when |x−a|<R and will diverge for when |x−a|>R.

Note that the series may or may not converge if |x−a|=R. What happens at these points will not change the radius of convergence.

Question 10

What is the interval of convergence?

Answer 10

The interval of all x’s, including the endpoints if need be, for which the power series converges is called the interval of convergence of the series.

If we know that the radius of convergence of a power series is R then we have the following.

a−R < x < a+R ==> power series converges

x < a−R & x > a+R ==> power series diverges

Question 11

How do you calculate the interval of convergence?

Answer 11

The interval of convergence must then contain the interval a−R < x < a+R since we know that the power series will converge for these values.

We also know that the interval of convergence can’t contain x’s in the ranges xv< a−R and x > a+R since we know the power series diverges for these values of x.

Therefore, to completely identify the interval of convergence all that we have to do is determine if the power series will converge for x = a−R or x = a+R.

If the power series converges for one or both of these values then we’ll need to include those in the interval of convergence.

Question 12

What is the shape of the region a power series converges to?

Answer 12

It is always a disc.

Question 13

Given a radius of convergence R = 0, for what values of x does a power series converge/diverge?

Answer 13

When a power series has convergence radius R = 0, it only converges for x = 0 and diverges for all other values of x.

Question 14

Given a radius of convergence R = r, such as in the geometric series, what values of x do such a series converge/diverge?

Answer 14

When the radius of convergence is R = r, it means that any R ϵ (0, ∞) can be the radius of convergence of a power series. Therefore, in theory, any value of x can be convergent, as it solely depends on R.

Question 15

Given a radius of convergence R = r, such as in the geometric series, what values of x do such a series converge/diverge?

Answer 15

When the radius of convergence is R = r, it means that any R ϵ (0, ∞) can be the radius of convergence of a power series. Therefore, in theory, any value of x can be convergent, as it solely depends on R.

Question 16

What does the Cauchy-Hadamard theorem tell us about particular values of x?

Answer 16

The Cauchy-Hadarmard theorem tells us that if a power series converges for some particular x_1, then it converges for all complex values of x within a set such that |x| < |x_1|.

Likewise, if the series diverges for some complex value x_2, then it diverges for all complex values of x with |x| > |x_2|.

The radius of convergence R must then satisfy the inequality |x_1| <= R <= |x_2|. The graph of which looks like a doughnut.

Question 17

What does the Cauchy-Hadamard theorem tell us about convergence when |x| = R?

Answer 17

Just from the Cauchy-Hadarmad theorem, it is impossible to tell whether or not there is convergence if |x| = R. In fact there are many possibilities:

1. Divergence for all complex x with |x| = R. E.g. the geometric series converges if and only if |x| < 1. Hence R=1 and the series diverges in particular whenever |x| = 1.
2. Convergence for some and divergence for other complex x with |x| = R. Take the harmonic series (x = 1) and its alternating variant (x = -1). The harmonic series diverges and the alternating harmonic converges.
3. Absolute convergence for all complex x with |x| = R.

Question 18

What is an alternative formula to compute the radius of convergence of a power series, using the ratio test instead of the root test?

Answer 18

Given a general power series, if the limit

r = lim(n->∞) |a_(n+1)| / |a_n|

exists, then R = 1/r is the radius of convergence using the ratio test.

Which converges absolutely if |x|r < 1 (|x|< 1/r), and diverges if |x|r > 1 (|x| > 1/r).

Question 19

When are both radius of convergence tests (ratio and root variants) useful?

Answer 19

The ratio test version is useful for many examples where the root test is difficult to apply. Namely, when there are factorials or large numbers present. Whereas, the root test is useful when there are n-th powers.

The ratio test version is less general though because it requires the limit to exist, whereas the limit superior always exists

Question 20

What are power expansions?

Answer 20

A series expansion is a representation of a particular function as a sum of powers in one of its variables, or by a sum of powers of another (usually elementary) function.

Question 21

Who is Abel?

Answer 21

Niels Abel was a Norwegian mathematician who proved the impossibility of solving algebraically the general equation of the fifth degree.

Cauchy and Abel were crucial to the founding of modern-day complex analysis.

Question 22

What is Abel’s theorem?

Answer 22

Abel’s theorem says that if a power series converges on (−1,1) and also at x = 1 then its value at x = 1 is determined by continuity from the left of 1.

Abel’s theorem guarantees that, if a real power series converges for some positive value of the argument, the domain of uniform convergence extends at least up to and including this point. Furthermore, the continuity of the sum function extends at least up to and including this point.

In layman’s terms, if you take complex value |z|, it is going to be absolutely convergent as long as it is < |α|, where α = R. The corollary is also true.

Question 23

What is a limit superior?

Answer 23

Given a bounded sequence (x_n), then the limit superior is the largest accumulation point or simply the largest limit. If the sequence is unbounded then the limit superior would be positive infinity.