Re: [分析] 初微(10)

[Dirichlet ( )]

※ 引述《PttFund (批踢踢基金)》之銘言：
:            ∞                      2 sin a
: Show that  Σ (1/2^n)(cos na) = -------------.
:           n=1                    5 - 4 cos a

e^ina     cos(na)       sin(na)
----- =  --------- + i ---------
 2^n        2^n           2^n

令 (e^ia)/2 = r

 N
Σ  r^n = r(1-r^n)/(1-r) -> r/(1-r) as N -> oo
n=1
             e^ia        cos(a) + isin(a)      2cos(a) - 1 + i2sin(a)
r/(1-r) = --------- = ---------------------- = ----------------------
           2 - e^ia    2 - cos(a) - isin(a)         5 - 4cos(a)

   oo                    2cos(a) - 1
故 Σ (1/2^n)(cos na) = -------------
   n=1                   5 - 4cos(a)

   oo                      2sin(a)
   Σ (1/2^n)(sin na) = -------------
   n=1                   5 - 4cos(a)

題目寫反了 XD

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