Convergence and Divergence – Introduction to Series
Convergence and Divergence – Introduction to Series

TESTS FOR CONVERGENCESuppose you are given and infinite sequence <strong>of</strong> terms a 0 , a 1 , a 2 , . . . The notation ∑ ∞n=0 a n tells you to addall these terms, as follows: Form the partial sumsS 0 = a 0S 1 = a 0 + a 1.S N = a 0 + a 1 + · · · + a N =and then take the limit <strong>of</strong> these sums as N → ∞. If this limit exists and is a number, then ∑ ∞n=0 a n iscalled a convergent infinite series. If the limit does not exist, or is ±∞, the series is said to diverge.WARNING:(i)(ii)There are two sequences here:The terms <strong>of</strong> the series, a 0 , a 1 , a 2 , . . . , andthe partial sums S 0 , S 1 , S 2 , . . . , formed by adding more and more <strong>of</strong> those terms.With infinite series, it is the limit <strong>of</strong> the partial sums that concerns us, but to decide if a series convergesor diverges, i.e. to decide whether this limit exists, we will <strong>of</strong>ten isolate and look at the terms <strong>of</strong> the seriesas a sequence only, rather than their sum. It is important that you understand this distinction.OPERATIONS ON SERIES∞∑∞∑1. c a n = c a n , for any constant c.n=02. If bothn=0∞∑a n andn=0∞∑b n converge, thenn=0Note: There are no formulas for∞∑a n b n orn=0∞∑(a n + b n ) =n=0∞∑n=0a nb n.n=0N∑n=0a n∞∑ ∞∑a n + b n .IMPORTANT SERIES∞∑1. Geometric: ar n =a , where a is any constant and −1 < r < 1. If r is outside <strong>of</strong> this1 − rn=0interval, the geometric series diverges. Try to understand the pro<strong>of</strong> <strong>of</strong> these statements. It will help youunderstand what partial sums are, how to take their limit, and it will make you remember what theirformula is in this particular case.∑2. Harmonic and Alternating Harmonic: ∞n=1 1 n , ∑ ∞n=1 (−1)n 1 n. The harmonic series diverges(integral test), while the alternating harmonic series converges (alternating series test).∞∑ 13. p-series: , where p is a constant. This series converges if p > 1 and diverges if p ≤ 1. Notenp n=1that the harmonic series is a p-series with p = 1.n=0

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