Kenneth Kuttler

162 CHAPTER 6. INFINITE SERIES

6.5 Convergence Because of CancellationSo far, the tests for convergence have been applied to non negative terms only. Sometimes,a series converges, not because the terms of the series get small fast enough, but because ofcancellation taking place between positive and negative terms. A discussion of this involvessome simple algebra and yields a much more subtle test for convergence.

Let {an} and {bn} be sequences and let An ≡ ∑nk=1 ak, A−1 ≡ A0 ≡ 0. Then if p < q

∑n=p

anbn =q

∑n=p

bn (An −An−1) =q

∑n=p

bnAn −q

∑n=p

bnAn−1

∑n=p

bnAn −q−1

∑n=p−1

bn+1An = bqAq −bpAp−1 +q−1

∑n=p

An (bn −bn+1) (6.7)

This formula is called the partial summation formula. It is just like integration by parts.This yields Dirichlet’s test 1.

Theorem 6.5.1 (Dirichlet’s test) Suppose An ≡ ∑nk=1 ak is bounded independent of

n, meaning for some C > 0, |An| ≤C for all n. and limn→∞ bn = 0, with bn ≥ bn+1 for alln. Then ∑anbn converges. Thus it makes perfect sense to write ∑

∞n=1 anbn.

Proof: This follows quickly from Theorem 6.1.8. Indeed, letting |An| ≤ C, and usingthe partial summation formula above along with the assumption that the bn are decreasing,∣∣∣∣∣ q

∑n=p

anbn

∣∣∣∣∣=∣∣∣∣∣bqAq −bpAp−1 +

q−1

∑n=p

An (bn −bn+1)

∣∣∣∣∣≤C

(∣∣bq∣∣+ ∣∣bp

∣∣)+Cq−1

∑n=p

(bn −bn+1) =C(∣∣bq

∣∣+ ∣∣bp∣∣)+C (bp −bq)

and by assumption, this last expression is small whenever p and q are sufficiently large.Thus the partial sums are a Cauchy sequence.

Definition 6.5.2 If bn > 0 for all n, a series of the form ∑k (−1)k bk or ∑k (−1)k−1 bkis known as an alternating series.

The following corollary is known as the alternating series test.

Corollary 6.5.3 (alternating series test) If limn→∞ bn = 0, with bn ≥ bn+1, then it fol-lows that the series ∑

∞n=1 (−1)n bn converges.

Proof: Let an = (−1)n . Then the partial sums of ∑n an are bounded and so Theorem6.5.1 applies.

In the situation of Corollary 6.5.3 there is a convenient error estimate available.

1Peter Gustav Lejeune Dirichlet, 1805-1859 was a German mathematician who did fundamental work inanalytic number theory. He also gave the first proof that Fourier series tend to converge to the mid-point of thejump of the function. He is a very important figure in the development of analysis in the nineteenth century. Aninteresting personal fact is that the great composer Felix Mendelsson was his brother in law.

162 CHAPTER 6. INFINITE SERIES6.5 Convergence Because of CancellationSo far, the tests for convergence have been applied to non negative terms only. Sometimes,a series converges, not because the terms of the series get small fast enough, but because ofcancellation taking place between positive and negative terms. A discussion of this involvessome simple algebra and yields a much more subtle test for convergence.Let {a,} and {b,} be sequences and let Ay = Y%_) ax, A-1 = Ao = 0. Then if p < qq q q qy Anby = y bn (An —Apn-1) = y byAn— y byAn-1n=p n=p n=p n=pq q-1 q-l= y byAn _ y? bn+iAn = bgAg _ bpAp-1 + y? An (bn _ bn+1) (6.7)n=p n=p-1 n=pThis formula is called the partial summation formula. It is just like integration by parts.This yields Dirichlet’s test !.Theorem 6.5.1 (Dirichlet’s test) Suppose An = Vy_, a is bounded independent ofn, meaning for some C > 0, |An| <C for all n. and Vimy. by = 0, with by > by+1 for alln. Then Yanby converges. Thus it makes perfect sense to write Vy 4 AnDn.Proof: This follows quickly from Theorem 6.1.8. Indeed, letting |A,| < C, and usingthe partial summation formula above along with the assumption that the b, are decreasing,q q-lY? anbn| = |bgAg— bpAp-1 + Y) An (bn — Dnt)n=p n=pq-l<C(|bg|+|bp|) +€ YS (bn — busi) = C ([bq| + |Pp|) +€ (bp — bg)n=pand by assumption, this last expression is small whenever p and gq are sufficiently large.Thus the partial sums are a Cauchy sequence. §fDefinition 6.5.2 7b, > 0 for alln, aseries of the form, (—1)* by or Ly. (—1)* 1! byis known as an alternating series.The following corollary is known as the alternating series test.Corollary 6.5.3 (alternating series test) If \imy scooby = 0, with by > bn+1, then it fol-lows that the series Y*_, (—1)" by converges.Proof: Let a, = (—1)". Then the partial sums of Y,, a, are bounded and so Theorem6.5.1 applies.In the situation of Corollary 6.5.3 there is a convenient error estimate available.'Peter Gustav Lejeune Dirichlet, 1805-1859 was a German mathematician who did fundamental work inanalytic number theory. He also gave the first proof that Fourier series tend to converge to the mid-point of thejump of the function. He is a very important figure in the development of analysis in the nineteenth century. Aninteresting personal fact is that the great composer Felix Mendelsson was his brother in law.