W10 - Examples

Ratio test and Root test

Ratio test examples

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

Notice this technique!

Simplify the ratio:

We frequently use these rules:

to simplify ratios having exponents and factorials.

(b) has ratio .

Simplify this:

So the series converges absolutely by the ratio test.

(c) Observe that has ratio as .

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

(d) Observe that has ratio as .

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

(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.

(b) Observe that has roots of terms:

Because , the RooT shows that the series converges absolutely.

(c) Observe that converges because as .

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.

Power series: Radius and Interval

Radius of convergence

Find the radius of convergence of the series:

(a) (b)

Solution

(a) The ratio of coefficients is .

Therefore and the series converges for .

(b) This power series has , meaning it skips all odd terms.

Instead of the standard ratio function, we take the ratio of successive even terms. The series of even terms has coefficients . So:

As , this converges to , so and .

Radius and interval for a few series

Series	Radius	Interval

Interval of convergence

Find the interval of convergence of the following series.

(a) (b)

Solution

(a)

1. Apply ratio test.
   • Ratio of successive coefficients:
   • Limit of ratios:
   • Deduce and therefore .
   • Therefore:
2. Preliminary interval of convergence.
   • Translate to interval notation:
3. Final interval of convergence.
   • Check endpoint :
   • Check endpoint :
   • Final interval of convergence:

(b)

1. Limit of coefficients ratio.
   • Ratio of successive coefficients:
   • Limit of ratios:
   • Deduce and thus .
   • Therefore:
   • Preliminary interval of convergence:
2. Check endpoints.
   • Check endpoint :
   • Check endpoint :
   • Final interval of convergence:

Interval of convergence - further examples

Find the interval of convergence of the following series.

(a) (b)

Solution

(a)

• Ratio of coefficients: .
• So the , center is , and the preliminary interval is .
• Check endpoints: diverges and also diverges. Final interval is .

(b)

• Ratio of coefficients: .
• So , and the series converges when .
• Extract preliminary interval.
  • Divide by :
• Check endpoints: converges but diverges.
• Final interval of convergence: