Math 321 - April 09, 2021 25 Properties of Infinite Series

One can check for convergence of an infinite series by comparing it to another series whose convergence of divergence is known.

Theorem 25.1 (Comparison Test). Let (ak) and (bk) be sequences satisfying 0≤ak ≤bk for all k∈N. P∞ P∞ (i) If k=1bk converges, then so does k=1ak. P∞ P∞ (ii) If k=1ak diverges, then so dose k=1bk. The Comparison Test applies only to series with nonnegative terms, but one can apply it to series in which the terms are taken with absolute values and use the following theorem to handle series with possible negative terms, too. P∞ P∞ Theorem 25.2 (Absolute convergence test). If the series k=1|ak| converges, then so does the series k=1ak. The converse of the previous theorem is false, as illustrated by the alternating Harmonic series ∞ X 1 1 1 1 1 1 (−1)k+1 =1− + − + − +.... k 2 3 4 5 6 k=1 Taking absolute values of the terms in this series will produce the Harmonic series, which diverges, while the alternating series itself converges by the following general result on alternating series.

Theorem 25.3 (Alternating Series Test). Let (ak) be a sequence satisfying

(i) a1 ≥a2 ≥a3 ≥...ak ≥ak+1 ≥..., and

(ii) ak →0 as k→∞. P∞ k+1 Then the alternating series k=1(−1) ak converges. To distinguish between converging series, and those converging with absolute values, we introduce appropriate terminology. P∞ P∞ Definition 25.4. If k=1|ak| converges, then we say that the original series k=1ak converges absolutely. If, P∞ P∞ on the other hand, the series k=1ak converges, but the series of absolute values k=1|ak| diverges, then we P∞ say that the original series k=1ak converges conditionally. Absolute convergence is a stronger notion of convergence, as illustrated by the Absolute Convergence Test, and it even allows for rearrangements of the series without affecting its sum. P∞ P∞ P∞ Definition 25.5. Given a series k=1ak, the series k=1bk is called a rearrangement of k=1ak, if there exists a bijection f :N→N such that bf(k) =ak for all k∈N. Theorem 25.6. If a series converges absolutely, then any rearrangement of this series converges to the same sum. The situation is completely different with conditionally convergent series. Theorem 25.7 (Riemann rearrangement theorem). If a series converges conditionally, then for any S ∈R there exists a rearrangement of this series that converges to S, as well as a rearrangement that diverges. The following test, which essentially compares a given series to an appropriate geometric series can be useful when checking for convergence of a series. P∞ Theorem 25.8 (Ratio test). Let ak be a series with ak =06 , and suppose k=1 ak+1 lim =r. k→∞ ak P∞ (i) If r<1, then the series k=1ak converges absolutely. P∞ (ii) If r>1, then the series k=1ak diverges. The ratio test is inconclusive, if r=1.