→ gs86137:(ii)跟(iii)看不太懂意思耶 140.128.194.71 05/27 19:17
推 owlran:拿f(x)去跟ln|x|的泰勒比對，之後湊出 219.85.184.10 05/27 21:24
→ owlran:ln(1+x-9)，再對它微分得f'(x) 219.85.184.10 05/27 21:25
※ 引述《gs86137 (小飛魚)》之銘言：
: Find the interval of convergence of (i) f(x) , (ii) f'(x) , (iii) ∫f(x)dx
: ∞ (-1)^n+1‧(x-9)^n
: of the power series f(x) = Σ --------------------
: n=1 n
(i) f(x)
(-1)^n+1‧(x-9)^n
A(n)=--------------------
n
(-1)^n+2‧(x-9)^n+1
A(n+1)=--------------------
n+1
│A(n+1)│
lim │------│ = │x-9│< 1 收斂
n>∞ │ A(n) │
→ 8 < x < 10
∞ 1
x=8代入f(x)得Σ ─ 調和級數發散
n=1 n
∞ 1
x=10代入f(x)得Σ ─ (-1)^n 收斂為-ln2
n=1 n
∴ 8 < x ≦ 10
the interval of convergence (8,10]
(ii) f'(x)
(x-9)^2 (x-9)^3 (x-9)^4
f(x)=(x-9) - -------- + -------- - ---------- + ..........
2 3 4
=ln(1+x-9) 8 < x ≦ 10
1
f'(x)= ----- 8 < x ≦ 10
x-8
(iii)∫f(x)dx
∫f(x)dx=∫ln(x-8)dx
=xln│x-8│-x + c x屬於R
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