作者lifesoar (gold)
看板trans_math
标题Re: 幂级数
时间Thu May 27 16:56:40 2010
※ 引述《gs86137 (小飞鱼)》之铭言：
: 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 收敛
n>∞ │ A(n) │
→ 8 < x < 10
∞ 1
x=8代入f(x)得Σ ─ 调和级数发散
n=1 n
∞ 1
x=10代入f(x)得Σ ─ (-1)^n 收敛为-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属於R
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 118.161.135.250
1F：→ gs86137:(ii)跟(iii)看不太懂意思耶 140.128.194.71 05/27 19:17
2F：推 owlran:拿f(x)去跟ln|x|的泰勒比对，之後凑出 219.85.184.10 05/27 21:24
3F：→ owlran:ln(1+x-9)，再对它微分得f'(x) 219.85.184.10 05/27 21:25