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# 高等微積分證明請大師幫忙

Suppose An > (-1) for all n . By suitable applications of Taylor's theorem to the functions ㏒(1+x) or e^x,

show that ΣAn is absolutely convergent if and only if Σ㏒(1+An) is absolutely convergent.

這題好難喔= =

我想不出來~

我想問這題證明啊

一定要先證明

if |x|<0.5, then 0.75|x| < | ln(1+x) | < 2|x|

然後利用這個結果才推得出來嗎?

### 2 Answers

- 天助Lv 71 decade agoFavorite Answer
preliminary: if |x|<0.5, then 0.75|x| < | ln(1+x) | < 2|x|.

pf. omit.(by Calculus)

Σ|A(n)| abs. conv. then lim(n->inf) |A(n)|=0

so, there exists N0>0, such that, if n>=N0, then |A(n)|< 0.5

thus | ln(1+A(n)) | < 2|A(n)| (by preliminary)

so Σ|A(n)| abs. conv. then Σ | ln(1+A(n)) | abs. conv. (by comparison)

Σ| ln(1+A(n) | abs. conv. then lim(n->inf) ln(1+A(n)| =0, lim(n->inf) A(n)=0

so, there exists N0>0, such that, n>=N0, then |A(n)|<0.5

thus | ln(1+A(n)) | > 0.75|A(n)|, or |A(n)| <(4/3) | ln(1+A(n)) |

so, Σ | ln(1+A(n)) | abs. conv. then Σ|A(n)| conv. (by comparison)

2010-05-27 00:33:36 補充：

因担心Σ ln(1+A(n)) 會加總至∞或-∞,so, 找一個上下界限制之

加絕對值是必須的,因comparison只討論正項級數