Convergence and Divergence

Convergence and Divergence

If we add infinitely, we may converge to a number, which is pretty crazy (we add infinitely, but still come to a number).

Definition

An infinite series of constants is defined as the limit of finite series:

Theorem

Given the series

If exists and equals some number L, we say the series "converges". Otherwise, it diverges.

Example

For example, a geometric series may converge.

Example

Does converge?

solution
The series is equivalent to , and the inside of the brackets diverges to infinity, so the whole series diverges.

Convergence Tests

Abstract

  • #Geometric Series
    • Given :
      If then the series converges, otherwise it diverges
  • #Divergence Test (aka Nth Term Test)
    • If , then the series diverges, otherwise we do not know anything
  • #Integral Test
    • Conditions:
    • Given and a function which is continuous, positive, and decreasing on with :
      For all , the series converges if and only if also converges.
  • #P-Series Test
    • Given :
      The series converges if and only if
  • #Comparison Test
    • Conditions:
    • Given with , if we can identity another sum such that:
      • for all , if converges, then also converges
      • for all , if diverges, then also diverges
  • #Limit Comparison Test (LCT)
    • Conditions:
    • If and , then and converge of diverge together.
  • #Alternating Series Test (AST, Leibniz Test)
    • Conditions: , the series is alternating between + and -
    • Given :
      If and the sequence is decreasing, the series converges
    • #Alternating Series Estimation Theorem (ASET)
      • Given convergent series , the error is bounded by
    • #Absolute and Conditional Convergence
      • If converges, then is absolutely divergent
      • If diverges, but converges, then is conditionally divergent
    • #Riemann Rearrangement Theorem (Riemann Series Theorem)
      • We can rearrange the order we add a conditionally convergent series to change it's resultant sum
  • #Ratio Test
    • Very OP
    • Suppose . Then:
      • If , the series is absolutely convergent (this is related to the geometric series)
      • If , we make no conclusion
      • If , the series is divergent
  • #Root Test
    • Rare. Ratio test expect with

"User Guide"

Geometric Series

Recall the formula for the sum of a geometric series:

Because is an explicit function of , we can use Limit Laws to evaluate:

For what does exist?

So our limit exists for sure for .

Theorem

converges if , and is divergent otherwise.

Example

Does converge?

solution
Since , the series diverges.

Example

Does converge?

solution
Since , the series converges.

Example

Does converge?

solution

but , so the series diverges.

Example

Does converge?

solution
Lets write it in summation notation:

Does it converge? Yes, only for .

What does it converge to?

Note we got the formula of (1) from .

Divergence Test (aka Nth Term Test)

In the case of a geometric series, for , we will almost never have an explicit expression for (i.e we cannot use limits), so usually we will look at the sum and infer the convergence.

Theorem

If , then diverges.

Warning

This theorem is not if and only if. Just because does not mean converges.

Example

Does converge?

solution

so the series diverges.

Example

Does converge?

solution

so the series diverges.

Example

Does converge?

solution
, and , so we cannot make a conclusion using the divergence test.

Integral Test

Applicable only to series with for all .

(see integral)

Theorem

Given and a function which is continuous, positive, and decreasing on with :
For all , converges if and only if converges.

Remark

Recall from Curve Sketching and Optimization that we can check if a function is decreasing by taking its derivative, if we cannot tell by inspection.

Intuition

Recall that an integral is the area under a curve. As such, if the area under a curve is finite, then the curve must also be finite.

Example

Does the series converge?

solution
Notice that if we try the divergence test, , so we cannot conclude anything.

But , and is positive, continuous, and decreasing.

(see improper integral)

since the integral diverges, so does the series.

Example

Does converge?

solution

Evaluating this using the Method of Substitution, .

Since the integral converges, the series also converges.

Example

Does converge?

solution

The integral diverges, so the series also diverges.

P-Series Test

Theorem

The series:

converges if and only if

Remark

Since this is an if and only is theorem, the series will diverge if .

Example

Does converge?

solution
, so it diverges

Example

Does converge?

solution
, so it converges

Comparison Test

Applicable only to series with for all .

Theorem

Given the series , for all , if we can identify another sum such that:

  • for all , if converges, then also converges
  • for all , if diverges, then also diverges
Intuition

Think squeeze theorem vibes. If and it converges, then "pushes" down to convergence. If and it diverges, then "pushes" up to divergence.

Example

Does converge?

solution
Compare this with . , but diverges, so the series also diverges.

Example

Does converge?

solution
We know converges by p-series. Since for all , by comparison test, the series converges.

Limit Comparison Test (LCT)

Applicable only to series with for all .

Theorem

If and , then and converge diverge together.

Example

Does converge?

solution
A comparison test won't work: .

But using limit comparison test:

We have , , so converges, because converges.

Example

Does converge?

solution
Compare this with .

so the sum converges by limit comparison test because also converges.

Example

Does the series converge?

solution
Compare this with . By p-series test, this diverges.

By limit comparison test, since the series diverges.

Remark

Use the limit comparison test first, then try the comparison test

Alternating Series Test (AST, Leibniz Test)

Applicable only to series with for all .

Theorem

Consider the series with for all . If and the sequence is decreasing (i.e ), then the series converges.

Example

Does the series converge?

solution
This is not a p-series, since there are negative terms. However, it is alternating, and is a decreasing function. Since , the series converges.

Example

Does the series converge?

solution
The series alternates, and is a decreasing function. Since , the series converges.

Alternating Series Estimation Theorem (ASET)

Theorem

Given a convergent alternating series , if we use the partial sum as an estimate of the sum . the error satisfies the inequality .

That is, if you stop after terms, the error is bounded by .

Intuition

This means the truncation error is less than the first term omitted. In other words, the next term will "overstep" the error, so we can use it as an upper bound.

"Proof"

.

Example

Given the Taylor Series
What is the error in ?

solution
This is an alternating series, so the error is bounded by the first omitted term:

Absolute and Conditional Convergence

On the one hand, alternating series are easy to handle, but there is a strange feature.

Definition

If converges, then is absolutely convergent.
If diverges, but converges, then is conditionally convergent

Theorem

A series is conditionally convergent if and only if it is convergent, the series of its positive terms diverges to infinity, and the series of its negative terms diverges to infinity.

Remark

All non-alternating series are absolutely divergent.

Example

Is the series conditionally of absolutely divergent?

solution
This is an alternating series and the terms are decreasing.

But diverges, so the series is conditionally convergent.

Aside: This is the Taylor series of , and this converges to . As we will see later, this isn't as straightforward as we would like.

Example

Is the series conditionally or absolutely divergent?

solution
This is an alternating series and the terms are decreasing.
The series converges (by p-series), so the series is absolutely convergent.

We are after absolute convergent series, because they behave like normal functions. Conditional convergence is fiction.

Riemann Rearrangement Theorem (Riemann Series Theorem)

What the sum converges to is conditional on the order the terms are added⁉️⁉️

Theorem

Suppose the sum is conditionally divergent. Let . Then there exists a permutation such that:

There also exists such that:

That is, if an infinite series is conditionally convergent, then its terms can be arranged in a permutation such that the new series converges to any arbitrary real number, or diverges.

Example

Recall the conditionally divergent series:

We can change the order of terms to make the sum anything we want.

Suppose I want the sum to equal :

Which is an alternating series which converges to .

Remark

Why? We can keep infinitely alternating between below 1.5 and above 1.5. Think about it like a scale. One one side are the odd terms (positive) and on the other are the even terms (negative). We have infinity on both sides of the scale. Since , It's not well-defined, and can be anything we want.

Conditional convergent series are a lie.

They are conditional on the order the terms are added.

There's some deity with an infinite number of weights to balance the scale to whatever they want.
- Prof. Matthew Scott

Ratio Test

This test is very powerful

Theorem

Suppose . Then:

  • If , the series is absolutely convergent
  • If , the test fails (we make no conclusion)
  • If , the series is divergent
Intuition

This tests tries to see if "behaves like" a geometric series when approaches infinity.

The ratio test on gives . Recall a geometric series only converges if .

Example

Does the series converge?

solution
We could use AST, but the ratio test is easier:

so the series is absolutely convergent.

Example

Does the series converge?

solution

so the series is divergent.

Example

For what values of does absolutely converge?

solution

so the series converges absolutely for all .

Note that this is the Taylor polynomial for .

Example

For what values of does absolutely converge?

solution

so the series converges absolutely for .

Note that this is the Taylor polynomial for .

Root Test

Same as ratio test with .

Theorem

Suppose . Then:

  • If , the series is absolutely convergent
  • If , the test fails (we make no conclusion)
  • If , the series is divergent