Calculus II
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W08 - Notes

Convergence

01 Theory

Theory 2

Monotone sequences

A sequence is called monotone increasing if for every .

A sequence is called monotone decreasing if for every .

In this context, ‘monotone’ just means it preserves the increasing or decreasing modality for all terms.

Monotonicity Theorem

If a sequence is monotone increasing, and bounded above by , then it must converge to some limit , and .

If a sequence is monotone decreasing, and bounded below by , then it must converge to some limit , and .

Terminology:

  • Bounded above by means that for every
  • Bounded below by means that for every

Notice!

The Monotonicity Theorem says that a limit exists, but it does not provide the limit value.

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02 Illustration

Example - Monotonicity Theorem

Monotonicity

Show that converges.

Solution

(1) Observe that for all .

Because , we know .

Therefore

(2) Change to and show is decreasing.

New formula: considered as a differentiable function.

Take derivative to show decreasing.

Derivative of :

(3) Simplify:

Denominator is . Numerator is . So and is monotone decreasing.

Therefore is monotone decreasing as .

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03 Theory

Theory 3

Series convergence

We say that a series converges when its partial sum sequence converges:

Let us apply this to the geometric series. Recall our formula for the partial sums:

Rewrite this formula:

Now take the limit as :

So we see that converges exactly when . It converges to .

(If then the denominator is , and if then the factor does not converge.)

Furthermore, we have the limit value:

This result confirms the formula we derived for the total for a geometric series. This time we did not start by assuming exists, on the contrary we proved that exists. (Provided that .)

Extra - Aspects of and from the geometric series

Notice that we always have the rule:

This rule can be viewed as coming from partitioning the full series into a finite part and the remaining infinite part:

We can remove a factor from the infinite part:

The parenthetical expression is equal to , so we have the formula given above.

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Simple divergence test

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Videos, Math Dr. Bob

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04 Theory

Theory 1

Simple Divergence Test (SDT)

Applicability: Any series.

Test Statement:

AKA the “Not Even Close” test

The converse is not valid. For example, diverges even though .

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05 Illustration

Example - Simple Divergence Test: examples

Simple divergence test: examples

(a) Consider:

This diverges by the SDT because and not .

(b) Consider:

This diverges by the SDT because .

We can say the terms “converge to the pattern ,” but that is not a limit value.

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Positive series

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06 Theory

Theory 1

Positive series

A series is called positive when its individual terms are positive, i.e. for all .

The partial sum sequence is monotone increasing for a positive series.

By the monotonicity test for convergence of sequences, therefore converges whenever it is bounded above. If is not bounded above, then diverges to .

Another test, called the integral test, studies the terms of a series as if they represent rectangles with upper corner pinned to the graph of a continuous function.

To apply the test, we must convert the integer index variable in into a continuous variable . This is easy when we have a formula for (provided it doesn’t contain factorials or other elements dependent on integrality).

Integral Test (IT)

Applicability: must be:

  • Continuous
  • Positive
  • Monotone decreasing

Test Statement:

Extra - Integral test: explanation

To show that integral convergence implies series convergence, consider the diagram:

This shows that for any . Therefore, if converges, then is bounded (independent of ) and so is bounded by that inequality. But ; so by adding to the bound, we see that itself is bounded, which implies that converges.

To show that integral divergence implies series divergence, consider a similar diagram:

This shows that for any . Therefore, if diverges, then goes to as , and so goes to as well. So diverges.

Notice: the picture shows entirely above (or below) the rectangles. This depends upon being monotone decreasing, as well as . (This explains the applicability conditions.)

Next we use the integral test to evaluate the family of -series, and later we can use -series in comparison tests without repeating the work of the integral test.

-series

A -series is a series of this form:

Convergence properties:

Extra - Proof of -series convergence

(1) To verify the convergence properties of -series, apply the integral test:

Applicability: verify it’s continuous, positive, decreasing.

Convert to to obtain the function .

Indeed is continuous and positive and decreasing as increases.

(2) Apply the integral test.

Integrate, assuming :

When we have

When we have

When , integrate a second time:

(3) Conclude: the integral converges when and diverges when .

Supplement: we could instead immediately refer to the convergence results for -integrals instead of reproving them here.

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07 Illustration

Example - -series examples

p-series examples

By finding and applying the -series convergence properties:

We see that converges: so

But diverges: so

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Example - Integral test - pushing the envelope of convergence

Integral test - pushing the envelope of convergence

Does converge?

Does converge?

Notice that grows very slowly with , so is just a little smaller than for large , and similarly is just a little smaller still.

Solution

(1) The two series lead to the two functions and .

Check applicability.

Clearly and are both continuous, positive, decreasing functions on .

(2) Apply the integral test to .

Integrate :

Conclude: diverges.

(3) Apply the integral test to .

Integrate :

Conclude: converges.

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