Examples

Ratio test examples

(a) Observe that has ratio and thus . Therefore the RaT implies that this series converges.

Simplify the ratio:

Notice this technique! We frequently use these rules:

(To simplify ratios with exponents and factorials.)

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(b) has ratio .

Simplify this:

So the series converges absolutely by the ratio test.

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(c) Observe that has ratio as .

So the ratio test is inconclusive, even though this series fails the SDT and obviously diverges.

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(d) Observe that has ratio as .

So the ratio test is inconclusive, even though the series converges as a -series with .

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(e) More generally, the ratio test is usually inconclusive for rational functions; it is more effective to use LCT with a -series.

Root test examples

(a) Observe that has roots of terms:

Because , the RooT shows that the series converges absolutely.

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(b) Observe that has roots of terms:

Because , the RooT shows that the series converges absolutely.

Ratio test versus root test

Determine whether the series converges absolutely or conditionally or diverges.

Solution Before proceeding, rewrite somewhat the general term as .

Now we solve the problem first using the ratio test. By plugging in we see that

So for the ratio we have:

Therefore the series converges absolutely by the ratio test.

Now solve the problem again using the root test. We have for :

To compute the limit as we must use logarithmic limits and L’Hopital’s Rule. So, first take the log:

Then for the first term apply L’Hopital’s Rule:

So the first term goes to zero, and the second (constant) term is the value of the limit. So the log limit is , and the limit (before taking logs) must be (inverting the log using ) and this is . Since , the root test also shows that the series converges absolutely.