Question

20£®ÈçͼËùʾ£¬ÓÉÖ±Ïßx=a£¬x=a+1£¨a£¾0£©£¬y=x²¼°xÖáΧ³ÉµÄÇú±ßÌÝÐεÄÃæ»ý½éÓÚÏàӦС¾ØÐÎÓë´ó¾ØÐεÄÃæ»ýÖ®¼ä£¬¼´a²£¼${¡Ò}_{a}^{a+1}$x²dx£¼£¨a+1£©²£®Àà±ÈÖ®£¬?n¡ÊN^*£¬$\frac{1}{n+1}$+$\frac{1}{n+2}$+¡­+$\frac{1}{2n}$£¼A£¼$\frac{1}{n}$+$\frac{1}{n+1}$+¡­+$\frac{1}{2n-1}$ºã³ÉÁ¢£¬ÔòʵÊýAµÈÓÚ£¨¡¡¡¡£©

• A£® $\frac{1}{2}$
• B£® $\frac{3}{5}$
• C£® ln2
• D£® ln$\frac{5}{2}$

Answer 1

·ÖÎö ÁîA=A₁+A₂+A₃+¡­+A_n£¬¸ù¾Ý¶¨»ý·ÖµÄ¶¨ÒåµÃµ½£ºA₁=-lnn+ln£¨n+1£©£¬Í¬ÀíÇó³öA₂£¬A₃£¬¡­£¬A_nµÄÖµ£¬Ïà¼ÓÇó³ö¼´¿É£®

½â´ð ½â£ºÁîA=A₁+A₂+A₃+¡­+A_n£¬
ÓÉÌâÒâµÃ£º$\frac{1}{n+1}$£¼A₁£¼$\frac{1}{n}$£¬$\frac{1}{n+2}$£¼A₂£¼$\frac{1}{n+1}$£¬$\frac{1}{n+3}$£¼A₃£¼$\frac{1}{n+2}$£¬¡­£¬$\frac{1}{2n}$£¼A_n£¼$\frac{1}{2n-1}$£¬
¡àA₁=${¡Ò}_{n}^{n+1}$$\frac{1}{x}$dx=lnx|${\;}_{n}^{n+1}$=ln£¨n+1£©-lnn£¬
ͬÀí£ºA₂=-ln£¨n+1£©+ln£¨n+2£©£¬A₃=-ln£¨n+2£©+ln£¨n+3£©£¬¡­£¬A_n=-ln£¨2n-1£©+ln2n£¬
¡àA=A₁+A₂+A₃+¡­+A_n
=-lnn+ln£¨n+1£©-ln£¨n+1£©+ln£¨n+2£©-ln£¨n+2£©+ln£¨n+3£©-¡­-ln£¨2n-1£©+ln2n
=ln2n-lnn
=ln2£¬
¹ÊÑ¡£ºC£®

µãÆÀ ±¾Ì⿼²ìÁ˶¨»ý·ÖµÄ¼òµ¥Ó¦Ó㬸ù¾Ý¶¨»ý·ÖµÄ¶¨ÒåµÃµ½A₁£¬A₂£¬A₃£¬¡­£¬A_nµÄÖµÊǽâÌâµÄ¹Ø¼ü£¬±¾ÌâÊÇÒ»µÀÖеµÌ⣮