MAT 127: Calculus C, Fall 2009Course Summary IIIExtremely Important: sequences vs. series (do not mix them or their convergence/divergencetests up!!!); what it means for a sequence or series to converge or diverge; power seriesVery Important: convergence/divergence tests for series; radius and interval of convergence forpower series; differentiation, integration, and limits of functions via power series; Taylor seriesImportant: estimating infinite sums by finite sums; finding radius and interval of convergence ofpower series; determining Taylor series of functions related to standard ones; applications of powerseries to computing sums of seriesSeries (cont’d)(9) Like the 3 convergence/divergence tests of 8.3 (Integral Test, Compari son Test, and Limit Com-parison Test), the Ratio Test is intended for series with positive terms:if the sequence {an} has positive terms and• limn−→∞an+1an= L < 1, then the series∞Xn=1anconverges;• limn−→∞an+1an= L > 1 or an+1/an−→ ∞, then the series∞Xn=1andiverges.This test says nothing if an+1/an−→ 1. This is not surprising si ncean=1np=⇒ limn−→∞an+1an= 1,while the seriesPanconverges if and only if p >1. Thus, the ratio test is not suitable for series thatinvolve only powers of n (e.g. n3), and not something growing faster. On the other hand, it usuallyworks very well with series that involve n! and exponents of n (e.g. 3nor nn). For this reason, thisis the test used to determine the rad ius of convergence of power series in 8.5-8.8 (but other tests areusually required to determine convergence at the end points of the resulting interval). Factors of ndo not effect the value of L in the Ratio Test; for example, L is the same for all three series∞Xn=113n,∞Xn=1n3n,∞Xn=11n3n.In 8.6 and 8.7, this translates into the derivative and anti-derivative of a power series having the sameradius of convergence as the original series (but the interval of convergence may change slightly).Unlike the 3 tests of 8.3, the Ratio Test is completely a self-test: you do not have to guess a secondsequence {bn}, which is required for Compa riso n Test and Limit Comparison Test, or a functionf = f(x) such that f(n) = an, which is required for Integral Test (“guessing” the function f usu-ally involves replacing n by x if this makes sense (e.g. x! does not); while this is easy, determiningwhether the resulting integral is finite or not may be less so). The Ratio Test is a consequence ofthe Comparison Test applied to a geometric series with r =(L+1)/2.(10) The Alternating Series Test applies to a narrow, but important, set of series with terms ofdifferent signs:if limn−→∞an= 0, |an|> |an+1|, and the signs of analternate (an> 0 for every n odd andan<0 for every n even, or the other way around), then the series∞Xn=1anconvergesThe alternating-sign condition is typically exhibited by the presence of (−1)nor (−1)n−1=−(−1)n;however, make sure to also check the first two conditions before concluding that the series converges.Typical examples are the series like∞Xn=1(−1)nn,∞Xn=1(−1)n−1(ln n)2n;both converge by the AST. The Alternating Series Test is a convergence test only: it states thata series converges if it meets 3 conditions. It can never be used to conclude that a series diverges;in this sense, it is the opposite of the most important divergence test, which can never be used toconclude that a series converges. If the first condition in the Alternating Series Test is not satisfied,the series does indeed diverge, but by the most important divergence test. However, there are lots ofseries that fail either the second or third condition (or both), but still converge; for example, thereare convergent series with only positive terms, that decay to zero, but are not strictly decreasing, e.g.∞Xn=12 + (−1)nn2.The Alternating Series Test is a consequence of the definition of convergence for series (convergenceof the sequence of partial sums) and the Monotonic Sequence Theorem.(11) The substance of Absolute Convergence Test is that introducing some minus signs into a con-vergent series with positive terms does not ruin the convergence:if the series∞Xn=1|an| c onverges, then so does the series∞Xn=1anThis test is useful when the signs are random, as opposed to strictly alternating as required forthe Alternating Convergence Test. For example, the series∞Xn=1sin nn2converges by the ACT, becausethe series∞Xn=1sin nn2=∞Xn=1|sin n|n2converges since 0 ≤ |sin n|/n2≤ 1/n2and the series∞Xn=11n2converges (this argument uses 3 tests:Absolute Convergence, Comparison, and p-Series; Limit Comparison Test is less suitable in this2case). The Alternating Series Test cannot be applied in this case, because the signs of sin n do notalternate:sin 1, sin 2, sin 3 > 0, sin 4, sin 5, sin 6 < 0;while the signs usually come in triples, occasionally there are four consecutive terms with the samesign. While the Absolute Convergence Test is less stringent about the alternating sign condition thanthe Alternating Series Test, the former is not a substitute of the latter. While either test can beused to conclude that the series∞Xn=1(−1)nn2converges, only the Alternating Series Test is applicableto the series∞Xn=1(−1)nnbecause the series∞Xn=1(−1)nn=∞Xn=11ndoes not converge. Neither of the two tests directly implies that the series∞Xn=1sin nnconverges1. Asthe Alternating Series Test, the Absolute Convergence Test is a convergence test only; it can neverbe used to conclude that a series diverges. The Absolute Convergence Test is a consequence of theComparison Test and the addition rule for series.(12) The general Ratio Test stated in the book extends the above Ratio Test for series with positiveterms to series with arbitrary nonzero terms (so that an+1/anmakes sense):if an6=0 for all n (≥ some N), and• limn−→∞an+1an= L < 1, then the series∞Xn=1anconverges;• limn−→∞an+1an= L > 1 oran+1/an−→ ∞, then the series∞Xn=1andiverges.The first statement follows from the Absolute Convergence Test and the Ratio Test for series withpositive terms. As with the Ratio Test for series with positive terms, the second statement followsfrom the most important divergence theorem as i t implies that |an| tends to infinity and thus thesequence {an} does not converge to 0. Similar ly to the Ratio Test for