Math 347 Infinite Series: Definitions and Theorems A.J. Hildebrand Infinite Series: Definitions and Theorems Definitions P1 • Infinite series: An infinite series is a formal expression of the form k=1 ak, i.e., a sum over (countably) infinitely many real numbers.1 P1 Pn • Partial sums: The partial sums of a series k=1 ak, are defined as sn = k=1 ak, i.e., sn is the sum of the first n terms of the series. P1 • Convergence and divergence: Convergence of an infinite series k=1 ak is defined in terms of convergence of the sequence of its partial sums: 1 n X X ak converges () lim ak exists () fsng converges n!1 k=1 k=1 P1 The series k=1 ak is said to be divergent if it does not converge in the above sense, i.e., if the sequence of partial sums fsng does not converge. For a convergent series, its sum is defined as the limit of its partial sums: n X s = lim sn = lim ak n!1 n!1 k=1 P1 P1 • Absolute convergence: A series k=1 ak is said to be absolutely convergent if k=1 jakj con- verges. Theorems The following are some standard results about series that you should know. Except for the Alternating Series Test (whose proof is a bit tricky), all of these results can easily be proved using the above definition of convergence of series and results about infinite sequences (e.g, algebraic properties, Monotone Convergence Theorem, and Cauchy Criterion). Try to carefully work out each of these proofs. This is a great exercise in applying the definitions and theorems about sequences and series. P1 • Cauchy Criterion for infinite series: An infinite series k=1 ak converges if and only if it satisfies the Cauchy Criterion for infinite series: n X For every > 0 there exists N 2 N such that for any integers n > m ≥ N, ak < k=m+1 P1 • n-th term test: If k=1 ak converges, then limn!1 an = 0. • Algebraic properties: P1 P1 { Sum Property: If k=1 ak and k=1 bk converge with sums A and B respectively, then so does P1 k=1(ak + bk), with sum A + B. P1 P1 { Scaling: If k=1 ak converges with sum A, and c 2 R, then k=1(cak) converges, with sum cA. 1 We consider only series in which the terms ak are real numbers. 1 Math 347 Infinite Series: Definitions and Theorems A.J. Hildebrand P1 P1 • Comparison test: If jakj ≤ bk for all k 2 N and k=1 bk converges, then k=1 ak converges as well. More generally, if there exist constants C 2 R and N 2 N such that jakj ≤ Cbk holds for all k ≥ N, P1 P1 then the convergence of k=1 bk implies that of k=1 ak. • Absolute convergence implies convergence: An absolutely convergent series is convergent. 2 • Alternating series test: If the terms ak (i) alternate in sign (i.e., are negative for odd k and positive for even k, or vice versa), (ii) are nonincreasing in absolute value (i.e., satisfy jak+1j ≤ jakj P1 for all k 2 N), and (iii) have limit 0 as k ! 1, then the series k=1 ak converges. Two Classical Series The following are important examples of series that you should be familiar with. You should know how to prove the convergence resp. divergence of these series. P1 k • Geometric series with ratio r: k=0 r . This series converges if jrj < 1, with sum 1=(1 − r), and diverges if jrj ≥ 1. P1 1 • Harmonic series: k=1 k . This series diverges. Series versus Sequences By far the most common mistake in dealing with infinite series is to mix up a series with a sequence, and P1 apply theorems and properties of sequences fang to series n=1 an, or vice versa. A sequence is simply an infinite list of terms, a1; a2;::: , while a series is an infinite sum of terms, P1 a1 +a2 +a3 +··· , or n=1 an; Convergence of a series is defined in terms of the sequence of the corresponding partial sums sn = a1 + a2 + ··· + an; it is not equivalent to the convergence of the sequence of its terms fang. Here are some examples illustrating the difference. P1 • The sequence f1=ng converges to 0, whereas the series n=1 1=n is divergent. n P1 n • The sequence f1=2 g converges with limit 0, whereas the series n=0 1=2 converges, with the sum equal to 1=(1 − 1=2) = 2 (by the geometric series formula). • The product of two convergent sequences fang and fbng, i.e., the sequence fanbng, converges, but P1 P1 the analogous result does not hold for series: The convergence of the series n=1 an and n=1 bn P1 does not imply that of the series n=1 anbn. (For a counterexample consider an alternating series P1 n p such as n=1(−1) = n.) P1 • If the series n=1 an converges, then the sequence fang converges to 0 (by the n-th term test). The converse, however, is not true. (A counterexample is the harmonic series.) 2As mentioned, the proof of this result is somewhat tricky; you need to know the result, but not the proof. 2.