5 - Number Series 2.0

Question 1

What is the Harmonic Series? And what is its convergence?

Answer 1

^∞ Σ_k = 1/k

It is a divergent series

Question 2

What is the alternating harmonic series? And what is its convergence?

Answer 2

^∞ Σ_k = (-1)^(k+1) * 1/k

It is a convergent series

Question 3

What is the general theorem on the convergence of alternating series?

Answer 3

The Leibniz test

Question 4

What is the Leibniz test?

Answer 4

Let (a_k) be a non-negative decreasing sequence with a_k —> 0. Then the alternating series:

∞Σk = (-1)^k * a_k

converges. Moreover, a_0 - a_1 ≤ ∞Σ(k=0) = (-1)^k * a_k ≤ a_0

Question 5

What happens if infinitely many terms in a series are rearranged (changed in order)?

Answer 5

The limit of the series may change. Hence, in general, the “commutative law” is not valid for “infinite sums”.

By rearranging an infinite number of terms in a series, the sequence of partial sums is changed. The new series of partial sums have a different limit than the original sequence of partial sums.

Question 6

What does conditional convergence mean?

Answer 6

A series is said to be conditionally convergent iff (“if and only if”) it is convergent, the series of its positive terms diverges to positive infinity, and the series of its negative terms diverges to negative infinity.

In other words, a series has conditional convergence if it converges but different orderings do not converge to the same value.

Examples of conditionally convergent series include the alternating harmonic series and the logarithmic series.

Question 7

What does unconditional convergence mean?

Answer 7

A series is said to be unconditionally convergent iff (“if and only if”) every rearrangement leads to the same limit value.

Question 8

Can series be both conditionally and unconditionally convergent?

Answer 8

Yes and no. Unconditionally convergent series, a.k.a. absolutely convergent series, are also conditionally convergent. However, the converse is generally not true.

Question 9

What is absolute convergence?

Answer 9

It is the same as unconditional convergence. A series (a_k) is called absolutely convergent if

∞Σ(k=0) |a_k|

is a convergent sequence in {R}

Question 10

What is the boundedness of the partial sums?

Answer 10

A series ∞Σ(k=0) a_k converges if and only if the sequence of partial sums s_n = nΣ(k=0) a_k is bounded.

Question 11

What do the ratio and root tests both rely on?

Answer 11

A comparison with a geometric series and its convergence properties.

Question 12

What is the (Limit) Ratio Test?

Answer 12

Suppose we have a series ∑a_n. Define,

L=lim_(n→∞)∣∣ a_(n+1) / a_n ∣

Then,

1. If L<1, the series is absolutely convergent (and hence convergent).
2. If L>1, the series is divergent.
3. If L=1, the series may be divergent, conditionally convergent, or absolutely convergent.

Question 13

What is the series expression of the exponential function?

Answer 13

exp(x) := ∞∑(k=1) x^k / k!

This is the Taylor series of the exponential function e^x

Question 14

What is the (Limit) Root test?

Answer 14

Suppose that we have the series ∑an. Define,

L = lim(n→∞) |an|^1/n

Then,

1. If L<1, the series is absolutely convergent (and hence convergent).
2. If L>1, the series is divergent.
3. If L=1, the series may be divergent, conditionally convergent, or absolutely convergent.

Question 15

What are double series?

Answer 15

They are sums over an “infinite array” with infinite rows and columns. As below:

∞∑(j,k=0) x_jk

Question 16

In what order can double series sum be taken in?

Answer 16

There are several possibilities:

1. Sum by rows. First, take the sum of each row, then add them up:

∞∑(j=0) ( ∞∑(k=0) x_jk )

2. Sum by columns. First, take the sum of each column, then add them up:

∞∑(k=0) ( ∞∑(j=0) x_jk )

3. Given a bijection σ : {N} —> {N} x {N}, consider the sum

∞∑(i=0) x_σ(i)

Question 17

How do the alternating harmonic series and the harmonic series differ?

Answer 17

They are very similar only differing in size.

Question 18

What is the Cauchy Criterion?

Answer 18

A necessary and sufficient condition for a sequence S_i to converge. The Cauchy criterion is satisfied when, for all epsilon>0, there is a fixed number N such that |S_j-S_i|<epsilon for all i,j>N.

Question 19

What is the re-arrangement of an absolutely convergent series?

Answer 19

Every rearrangement of an absolutely convergent series converges (absolutely) to the same sum.

Question 20

What is a Cauchy Product?

Answer 20

The Cauchy product is the discrete convolution of two infinite series.

Given two absolutely convergent series (a_n) and (b_n). Then the series with terms equal to all possible products a_m*b_n, taken in any order, is also absolutely convergent.

Moreover,

∞∑(m,n) (a_m*b_n) = (∞∑n a_m) * (∞∑m b_n)

If two series converge absolutely than their Cauchy product converges absolutely to the inner product of the limits.

Question 21

What is a Cauchy Product?

Answer 21

The Cauchy product is the discrete convolution of two infinite series.

Given two absolutely convergent series (a_n) and (b_n). Then the series with terms equal to all possible products a_m*b_n, taken in any order, is also absolutely convergent.

Moreover,

∞∑(m,n) (a_m*b_n) = (∞∑n a_m) * (∞∑m b_n)

If two series converge absolutely then their Cauchy product converges absolutely to the inner product of the limits.

Question 22

Why use the Root test over the Ratio? Or vice versa?

Answer 22

While the Ratio Test is good to use with factorials, since there is that lovely cancellation of terms of factorials when you look at ratios, the root test is best used when there are terms to the n-th power with no factorials.

Question 23

What is the divergence test?

Answer 23

The divergence test says that if the If the terms of an infinite series don’t approach zero, the series must diverge.