Ngô Quốc Anh

October 18, 2007

Một số dạng bài tập xét sự hội tụ của chuỗi số dương

Filed under: Giải Tích 4 — Ngô Quốc Anh @ 14:54

1. Let \sum a_n be a convergent series, with a_n \geq 0. What can we say about the nature of each of the following series?

(a) \displaystyle\sum\sqrt {a_n} (b) \displaystyle\sum a_n^2 (c) \displaystyle\sum \frac {a_n}{1 + a_n}
(d) \displaystyle\sum \frac {a_n^2}{1 + a_n^2} (e) \displaystyle\sum \frac {\sqrt {a_n}}{n} (f) \displaystyle\sum \frac {a_n}{\sqrt {n}}
(g) \displaystyle\sum \frac {a_n}{n} (h) \displaystyle\sum na_n (i) \displaystyle\sum \cos(a_n)
(j) \displaystyle\sum \sin(a_n) (k) \displaystyle\sum \tan(a_n) (l) \displaystyle\sum \cos(\sqrt {a_n})
(m) \displaystyle\sum [1 - \cos(\sqrt {a_n})] (n) \displaystyle\sum [1 - \cos(a_n)] (o) \displaystyle\sum \arcsin(a_n)
(p) \displaystyle\sum \arccos(a_n) (q) \displaystyle\sum a_n^{3/2} (r) \displaystyle\sum a_n^3
(s) \displaystyle\sum \frac {a_1 + a_2 + ... + a_n}{n} (t) \displaystyle\sum \arctan(a_n) (u) \displaystyle\sum \frac {1}{1 + a_n}
(v) \displaystyle\sum ( - 1)^na_n (w) \displaystyle\sum \sqrt [n]{a_n} (x) \displaystyle\sum (a_n - a_{n + 1})
(y) \displaystyle\sum [\ln(1 + a_n)]^{5/4}

2) Regarding the same series of the previous question, how about if \sum a_n diverges?

Hint.

\displaystyle\left( a \right) \to a_n = \frac{1}{{n^2 }}.

\displaystyle\left( b \right) \to a_n^2 \leqslant a_n \sum\limits_{n = 0}^{ + \infty } {a_n } .

\displaystyle\begin{gathered}\left( d \right) \to \frac{{a_n^2 }}{{1 + a_n^2 }} \leqslant \frac{{a_n^2 }}{{2a_n }} = a_n \hfill \\\left( c \right) \to \frac{{a_n }}{{1 + a_n }} \leqslant a_n . \hfill \\\end{gathered}

\displaystyle\left( e \right) \to \frac{{\sqrt {a_n } }}{n} \leqslant \frac{1}{2}\left( {a_n + \frac{1}{{n^2 }}} \right)

\displaystyle\left( f \right) \to \mathop {\lim }\limits_{n \to \infty } \frac {{\frac {{a_n }} {{\sqrt n }}}} {{a_n }} = \mathop {\lim }\limits_{n \to \infty } \frac {1} {{\sqrt n }} = 0= \mathop {\lim }\limits_{n \to \infty } \frac {1} {n} = \mathop {\lim }\limits_{n \to \infty } \frac {{\frac {{a_n }} {n}}} {{a_n }} \leftarrow \left( g \right)

\displaystyle\left( h \right) \to a_n = \frac{1}{{n^2 }}.

\displaystyle\left( i \right) \to .

\displaystyle\left( j \right) \to |\sin(a_n)| \leqslant a_n.

\displaystyle\left( k \right) \to .

\displaystyle\left( l \right) \to .

\displaystyle\left( m \right) \to .

\displaystyle\left( n \right) \to .

\displaystyle\left( o \right) \to .

\displaystyle\left( p \right) \to .

\displaystyle\begin{gathered} \left( q \right) \to a_n^{\frac{3}{2}} \leqslant a_n \hfill \\\left( r \right) \to a_n^3 \leqslant a_n \hfill \\\end{gathered}

\displaystyle\left( s \right) \to \frac{{a_1 }}{n} \leqslant \frac{{a_1 + ... + a_n }}{n}.

\displaystyle\left( t \right) \to .

\displaystyle\left( u \right) \to a_n = \frac{1}{{n^2 }}.

\displaystyle\left( v \right) \to .

\displaystyle\left( w \right) \to a_n = \frac{1}{{n^n }}.

\displaystyle\left( x \right) \to |a_n-a_{n+1}| \leqslant a_n + a_{n+1}.

\displaystyle\left( y \right) \to .

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6 Comments »

  1. giai di. sao co moi bai tap ma khong co loi giai. thi ai biet dau ma lam ha

    Comment by giai — May 16, 2009 @ 15:54

    • Có gợi ý rồi mà :)

      Comment by Ngô Quốc Anh — December 16, 2009 @ 22:48

  2. giải thích rõ hơn đi bạn…

    Comment by tung hee — December 4, 2011 @ 21:04

    • Chào tung hee, đó là các dạng, đề bài phát biểu quá rõ ràng rồi, còn cần giải thích gì nữa.

      Comment by Ngô Quốc Anh — December 4, 2011 @ 21:14

  3. giai thich dum minh cau a @@

    Comment by quang — February 16, 2012 @ 22:28

    • Câu a) ta không thể khẳng định được chuỗi chưa căn hội tụ. Phản ví dụ như đề cập ở dưới, dùng \frac{1}{n^2}.

      Comment by Ngô Quốc Anh — February 16, 2012 @ 23:57


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