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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.

Theory 2

Direct Comparison Test (DCT)

Applicability: Both series are positive: and .

Test Statement: Suppose for large enough . (Meaning: for with some given .) Then:

  • Smaller pushes up bigger:
  • Bigger controls smaller:

Theory 3

Some series can be compared using the DCT after applying certain manipulations and tricks.

For example, consider the series . We suspect convergence because for large . But unfortunately, always, so we cannot apply the DCT.

We could make some ad hoc arguments that do use the DCT, eventually:

Trick Method 1:

  • Observe that for we have . (Check it!)
  • But converges, indeed its value is , which is .
  • So the series converges.

Trick Method 2:

  • Observe that we can change the letter to by starting the new at .
  • Then we have:
  • This last series has terms smaller than so the DCT with (a convergent -series) shows that the original series converges too.

These convoluted arguments suggest that a more general version of Comparison is possible.

Indeed, it is sufficient to compare the relative large-n behavior of the two series. We use the termwise ratios to estimate comparative behavior for increasing .

Limit Comparison Test (LCT)

Applicability: Both series are positive: and .

Test Statement: Suppose that . Then: If , i.e. finite non-zero, then:

Extra - LCT edge cases

If or , we can still draw an inference, but only in one direction:

  • If :
  • If :

Extra - Limit Comparison Test explanation

Suppose and . Then for sufficiently large, we know .

Doing some algebra, we get for large.

If converges, then also converges (constant multiple), and then the DCT implies that converges.

Conversely: we also know that , so for all sufficiently large. Thus if converges, also converges, and by the DCT again converges too.

The cases with or are handled similarly.