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Ma 116 Summary of Key Topics Spring, 2007 Sections 4.5, 5.10 L’Hospital’s Rule : Recognize indeterminate forms of the type 0 / 0, / , 0 · ∞ ; Convert to 0 / 0 or / before applying L’Hospital’s Rule; Then, lim x a f ( x ) g ( x ) = lim x a f 0 ( x ) g 0 ( x ) (if the limit on right-hand side exists) After applying L’Hospital’s rule, ﬁrst simplify the expression before determining if the new limit exists. For Indeteminate powers of the type 0 0 , 0 , 1 , ﬁrst apply the logarithm function and continuity to get, lim x a f ( x ) g ( x ) = exp h lim x a g ( x ) ln ( f ( x )) i and apply L’Hospital’s rule to the indeterminate form lim x a g ( x ) ln ( f ( x )) . Improper Integrals : Recognizing and evaluating improper integrals of Type I and II. Type I: integrals on inﬁnite intervals; deﬁned as a limit of deﬁnite integrals on ﬁnite intervals letting one endpoint tend to . Integrals covering the entire real line ( -∞ < x < ) must be split into two improper integrals on semi-inﬁnite intervals. Ex: Z -∞ x ( x 2 + 1 ) 2 / 3 dx = lim a →-∞ Z 0 a x ( x 2 + 1 ) 2 / 3 dx + lim b →+∞ Z b 0 x ( x 2 + 1 ) 2 / 3 dx Type II: integrand has a discontinuity in the interval of integration (typically a point where the integrand is not deﬁned); if necessary, break up the interval so the discontinuity appears at an endpoint; deﬁne the improper integral as a limit of deﬁnite integrals with one endpoint tending to the point of discontinuity. Ex: Z 3 0 1 ( x - 1 ) 2 dx = lim a 1 - Z a 0 1 ( x - 1 ) 2 dx + lim b 1 + Z 3 b 1 ( x - 1 ) 2 dx Chapter 8 Sequences : Deﬁnition of the limit of a sequence and determining convergence/divergence. Main Theorem on Sequences : every bounded, monotone sequence is convergent. Inﬁnite Series : Convergence/divergence is deﬁned with respect to the sequence of partial sums . N th partial sum: S N N X k = 1 a k and S = X k = 1 a k lim N →∞ S N

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Geometric Series : X n = 0 a r n = a + ar + ar 2 + . . . = a 1 - r for | r | < 1 and diverges for | r | ≥ 1. p -Series : X n = 0 1 n p converges for p > 1 and diverges for p 1. Absolute Convergence : X | a n | converges X a n converges. [ The converse is not true ] Convergence Tests : Typically begin with the ratio test for the series of absolute values. The other tests are necessary only if the ratio test is inconclusive (the limit equals 1). Ratio Test : Evaluate the limit lim n →∞ ± ± ± ± a n + 1 a n ± ± ± ± = L . L
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