Mathematics 116 Term Test or Test for Divergence

To apply the Term Test (for Divergence):

∞ Testing an for convergence or divergence. X

Investigate lim an. n →∞

∞ If lim an = 0, conclude that an diverges. n →∞ 6 X∞ If lim an = 0, conclude that an has a chance to converge but further testing is needed to n →∞ decide whether it does or not. TheX series an might converge or it might diverge. In sum, the Term Test give no information in this case.X

A formal look at why the Term Test is legitimate.

Theorem. If lim Sn = L for some number L, then lim an = 0. In other words, if the partial-sum n n sequence converges,→∞ then the term sequence must converge→∞ to 0.

Proof. The idea behind the proof is to find a way to write an by using the partial-sum sequence. Notice that an = Sn Sn 1. Now suppose that lim Sn = L (where L is a number, not or − n − →∞ ∞ ). Then lim Sn 1 = L, as well. Use the appropriate limit theorem for sequences we can write n − −∞ →∞ lim an = lim (Sn Sn 1) = lim Sn lim Sn 1 = L L = 0. n n − n n − →∞ →∞ − →∞ − →∞ −

By what’s often called contraposition, we can reformulate this theorem in a different way.

Corollary. If lim an = 0 (in other words if this limit either does not exist at all or is not zero), n →∞ 6 ∞ then an diverges. X

Proof. Contraposing the previous theorem tells us that if lim an = 0, then lim Sn cannot be a n n →∞ 6 →∞ ∞ number, and the definition of convergence for series then tells us that an diverges. X

Page 1 of 2 A.Sontag November 19, 2001 Term Test continued

An informal look at why the Term Test is legitimate

Suppose lim an = 3, say. This tells us that when n is huge, an 3. So n →∞ ≈ a = a + a + a + some number Q+ 3+ 3+ 3 + = n 1 2 3 · · · ≈ ≈ ≈ ≈ ··· ∞ XThis should convince us that if a 3, then S + and a diverges. n −→ n −→ ∞ n X

Practice Problems

What information, if any, does the Term Test give about the convergence or divergence of the following series?

∞ 1 1. 2n + 3 n=1 X

∞ 3n 2. 2n + 4 n=1 X

n ∞ 1 3. 1 + n n=1 X

∞ n2 4. en n=1 X

∞ 2n2 + 4n 5. 5n2 + 6n n=1 X

∞ 1 6. n ln n n=2 X ∞ n 7. ln n + 1 n=1 X Answers. 1. No information. 2. Series diverges. 3. Series diverges. 4. No information. 5. Series diverges. 6. No information. 7. No information.

Page 2 of 2 A.Sontag November 19, 2001