AP Calculus BC

Does the Sum of 3^n/n Converge? No

The sum of 3 to the n over n diverges. The ratio test decides it: the ratio of consecutive terms tends to 3, which is greater than 1. The terms themselves blow up, so the nth term test gives the same verdict in one line.

∑n=1∞3nn\sum_{n=1}^{\infty}\frac{3^{n}}{n}

Diverges

Settled by the ratio test.

The ratio test pointing the other way

Same setup as the convergent versions of this shape, with the exponential moved above the line.

an+1an=3n+1n+1⋅n3n=3⋅nn+1\frac{a_{n+1}}{a_{n}} = \frac{3^{n+1}}{n+1}\cdot\frac{n}{3^{n}} = 3\cdot\frac{n}{n+1}
L=lim⁡n→∞3⋅nn+1=3>1L = \lim_{n \to \infty} 3\cdot\frac{n}{n+1} = 3 > 1

L>1L > 1 is a divergence result, not an inconclusive one. The ratio test is undecided only at L=1L = 1.

Flipping the fraction flips the verdict

The series with terms n over 3 to the n converges, with ratio limit 1 over 3. The same two pieces in the opposite arrangement give the opposite answer, because the ratio limit turns into its reciprocal.

The shorter justification

The nth term test settles this faster. Powers of 33 outgrow nn, so the terms increase without bound.

lim⁡n→∞3nn=∞≠0\lim_{n \to \infty}\frac{3^{n}}{n} = \infty \neq 0

Under exam conditions, write the shortest correct argument. Two lines about the terms failing to reach zero score exactly what a full ratio test computation scores.

The mistakes students make

Divergence by ratio test is the case students most often mishandle.

  • Treating L>1L > 1 as inconclusive. Only L=1L = 1 leaves the question open.
  • Cancelling 3n+13n\frac{3^{n+1}}{3^{n}} down to 11 instead of 33. That leaves an+1an=nn+1\frac{a_{n+1}}{a_{n}} = \frac{n}{n+1}, whose limit is exactly 11, the one value at which the ratio test decides nothing. The slip does not reverse the verdict, it destroys the argument, and students then read nn+1<1\frac{n}{n+1} < 1 straight off the expression without taking the limit and announce convergence anyway.
  • Assuming an nn in the denominator drags the terms to zero. The numerator here is exponential and wins with room to spare.

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

Does the sum of 3^n/n converge?

No. The ratio test gives L=3>1L = 3 > 1, so the series diverges.

Which test is quickest for 3^n/n?

The nth term test, since the terms tend to ∞\infty. The ratio test reaches the same verdict with L=3L = 3.

What does L greater than 1 mean in the ratio test?

Divergence, decisively. The test fails to decide only when L=1L = 1.