Re: [試題] 100資工系轉系考題目(微積分部分)
※ 引述《weisl (好倫~)》之銘言:
: 2.
: ∞
: 求 Σ n/(2^n)
: n=1
1 ∞ n
── = Σ x
1-x n=0
兩邊微分
1 ∞ n-1
──── = Σ n x
(1-x)^2 n=1
∞ n 1 ∞ n 1 1
Σ ── = ─ Σ ──── = ─ ────
n=1 2^n 2 n=1 2^(n-1) 2 (1-1/2)^2
: 4. (100)
: f(x)=x^2/(x+1)(x+2)(x+3) , 求f (0)=?
1 4 9
f(x) = ──── - ─── + ────
2(x+1) (x+2) 2(x+3)
n n n n
1 ∞ n n ∞ (-1) x 3 ∞ (-1) x
= ─ Σ (-1) x - 2 Σ ── + ─ Σ ──
2 n=0 n=0 2^n 2 n=0 3^n
100 100
(100) 1 100 (-1) 1 (-1)
f (0) = 100! ( ─ (-1) - ─── + ─ ── )
2 2^99 2 3^99
: 6.
: lim (arctan x/x)^(1/x^2)=? //感謝wolf30強者更正
: x→0
1
──
arctanx x^2
lim ( ──── )
x→0 x
1
2 ──
x x^2
= lim ( 1 - ─ + ... )
x→0 3
-1/3
= e
--
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