Theory

Theory 1

Ratio Test (RaT)

Applicability: Any series with nonzero terms.

Test Statement:

Suppose that as .

Then:

Extra - Ratio Test - a deeper look

To understand the ratio test better, first consider this series:

• The term is created by multiplying the prior term by .
• The term is created by multiplying the prior term by .
• The term is created by multiplying the prior term by .

When , the multiplication factor giving the next term is necessarily less than . Therefore, when , the terms shrink faster than those of a geometric series having . Therefore this series converges.

Similarly, consider this series:

Write for the ratio from the prior term to the current term . For this series, .

This ratio falls below when , after which the terms necessarily shrink faster than those of a geometric series with . Therefore this series converges.

The main point of the discussion can be stated like this:

Whenever this is the case, then eventually the ratios are bounded below some , and the series terms are smaller than those of a converging geometric series.

Extra - Ratio Test proof

Let us write for the ratio to the next term from term .

Suppose that as , and that . This means: eventually the ratio of terms is close to ; so eventually it is less than .

More specifically, let us define . This is the point halfway between and . Since , we know that eventually .

Any geometric series with ratio converges. Set for big enough that . Then the terms of our series satisfy , and the series starting from is absolutely convergent by comparison to this geometric series.

(Note that the terms do not affect convergence.)

Theory 2

Root Test (RooT)

Applicability: Any series.

Test Statement:

Suppose that as .

Then:

Extra - Root test: explanation

The fact that and implies that eventually for all high enough . Set (the midpoint between and ).

Now, the equation is equivalent to the equation .

Therefore, eventually the terms are each less than the corresponding terms of this convergent geometric series: