相當(dāng)于求當(dāng)n→∞時(shí),A=3/[cos15°·cos(15°/2)·…·cos(15°/2^n)]的取值.
分子分母同時(shí)乘以 2^n·sin(15°/2^n),有
A= 3·2^n·sin(15°/2^n)/[cos15°·cos(15°/2)·…·2^n·sin(15°/2^n)·cos(15°/2^n)]
= 3·2^n·sin(15°/2^n)/【cos15°·cos(15°/2)·…·2^(n-1)·sin[15°/2^(n-1)]·cos[15°/2^(n-1)]】
= ……
= 3·2^n·sin(15°/2^n)/[cos15°·2·sin(15°/2)·cos(15°/2)]
= 3·2^n·sin(15°/2^n)/[cos15°sin15°]
= 3·2^n·sin(15°/2^n)/[1/2·2cos15°sin15°]
= 3·2^n·sin(15°/2^n)/[1/2·sin30°]
= 12·2^n·sin(15°/2^n)
取極限n→∞,由等價(jià)無窮小代換,有sin(15°/2^n)~15°/2^n
因此lim(n→∞)A= lim(n→∞) 12·2^n·sin(15°/2^n)
= lim(n→∞) 12·2^n·(15°/2^n)
= 12·π·15°/180°
= π