AP Calculus BC

Does the Sum of 4^n/5^n Converge? Yes, to 4

Q: Is a slowly shrinking geometric series still convergent?

Yes. Any ratio with |r| < 1 converges, however close to 1 it sits. The rate affects how many terms you need for a good numerical estimate, never the verdict.

Q: What if the ratio were 5/4 instead?

Then the terms would grow without bound, so they could not tend to 0, and the series would diverge by the nth term test before the geometric formula was ever needed.

The series converges to exactly 4. Two powers stacked in a quotient always collapse into one geometric ratio, here 4/5, and since that is less than one in absolute value the series converges.

\sum_{n=1}^{\infty}\frac{4^{n}}{5^{n}}

Converges

\text{sum} = 4

Settled by the geometric series test.

Collapse the powers first

A quotient of two nth powers is a single nth power: $\frac{4^{n}}{5^{n}} = \left(\frac{4}{5}\right)^{n}$ . Doing that first turns a series that looks unfamiliar into the most standard shape there is.

The ratio is $4/5 = 0.8$ . It is close to 1, so the terms shrink slowly and the partial sums take a long time to settle, but slow is not the same as divergent. The verdict turns on whether the ratio is less than 1, not on how much less.

The sum

\frac{4/5}{1 - 4/5} = \frac{4/5}{1/5} = 4

Notice how sensitive the sum is near the boundary. A ratio of $4/5$ gives 4, but $9/10$ would give 9 and $99/100$ would give 99. The sum runs to infinity as the ratio approaches 1, which is the boundary behaviour the geometric test is describing.

Not sure which test a series wants?

The Convergence Test Chooser walks the decision in order: nth term first, then geometric and p-series pattern matching, then alternating structure, then the ratio test, and finally the comparison family.

Frequently asked questions

Is a slowly shrinking geometric series still convergent?

Yes. Any ratio with $|r| < 1$ converges, however close to 1 it sits. The rate affects how many terms you need for a good numerical estimate, never the verdict.

What if the ratio were 5/4 instead?

Then the terms would grow without bound, so they could not tend to 0, and the series would diverge by the nth term test before the geometric formula was ever needed.