3.请叙述“无穷积分十ef(r)dr的Cauchy收敛准则.” 答：Cauchy收敛准则：无穷积分efr)dr收敛→ e>3A>a>Am>4有a恤< 4.请叙述“f(e)g(x)d的Dirichlet判别法." 答：Dirichlet判别法：设函数f)与g()在区间a,+o)有定义，在任何有穷 区间a.A都可积.若 四)积分F(A)=g)为A的有界函数，即K>0,A>a有1F(川 LA f(a)dx<K: ②)函数f回)是单调的，且1imf)=0, 则无穷积分∫+f(x)g(x)dx收敛. 数学分析四试题第2页（共8页）3. ➒◗ã“ ➹→➮➞ R +∞ a f(x)dx ✛ Cauchy ➶ñ❖❑.” ❽: Cauchy ➶ñ❖❑: ➹→➮➞ R +∞ a f(x)dx ➶ñ⇐⇒ ∀ > 0, ∃A > a, ∀p1 > A, ∀p2 > A, ❦     Z p2 p1 f(x)dx     < . 4. ➒◗ã“ R +∞ a f(x)g(x)dx ✛ Dirichlet ✞❖④.” ❽: Dirichlet ✞❖④: ✗➻ê f(x) ❺ g(x) ✸➠♠ [a, +∞) ❦➼➶, ✸❄Û❦→ ➠♠ [a, A] Ñ➀➮, ❡ (1) ➮➞ F(A) = R A a g(x)dx ➃ A ✛❦✳➻ê, ❂ ∃K > 0, ∀A > a ❦ |F(x)| =    R A a f(x)dx    ≤ K; (2) ➻ê f(x) ➫ü◆✛, ❹ lim x→+∞ f(x) = 0, ❑➹→➮➞ R +∞ a f(x)g(x)dx ➶ñ. ê➷➞Û(III)➪❑ ✶ 2 ➄↔✁ 8 ➄↕