��ƪ��s�q�n��

�Ҧ� (a,b] �� [a,b) �϶��u�L��j��ơv�s�q�n��A �g�L��ƪ��M��g��A�`�O�i�H�ܦ� (0,1] �϶��D�C �Ҧp�A�p�G f(x) �b (a,b] ��s��A�h g(x) = f(x+a) �N�ܦ��b (0,b-a] ��s��A��

�N�ܦ��b (0,1] �϶��s��C�M�� f(x) �b (a,b] �϶��n��A �N�ܦ� g(x) �b (0,1] �϶��n�� (�o��N�O�n��ܼ��ܴ��ޥ��Ff(x) �b (a,b] ��n�ȩM g(x) �b (0,1] ��n�ȡA�u�t�@�ӱ`�ƭ��ơC�i��̪��s�q�n��A �n�O��ĴN��̳��ġA�n�O�o��h��̳��o��C)

�Ҧ� �u�L��϶��v��s�q�n��D�A�]��i�H�g�L��A �ӧ令 �϶��s�q�n��C

�]��A�n��A�ʦa�Q�׼s�q�n��O�_��Ī��D�A �i�H�u�Q�צb (0,1] ��s��ӥB�� ɷ|�o��ơF�٦��b ��s�򪺨�ơC �өҦ��o��ơA�S�i�H²�ƨ쾭�� x^p �ӰQ�סC

�p�G�� p > 0 �h x^p �b (0,1] ��ڥ��O�@�ӥi�ݮi��s��ơA �ҥH��ڥ��O�s�q�n��A�u�O�Ӵ��q��n��A�ӥB�b [0,1] ��w�i�n�C �Ӧp�G�� p > 0 �h x^p �b ��n��w�o��A�o�O�]�� ҥH

��@�ӼƷ|�j��

��Ӳ@�L�ðݦa�A��ۤv�]�N�o��L�a�j�C

�ҥH�A��󥿪�� p�A�ڥ��S��ȱo�Q�ת��s�q�n��C ��A�{�b�ڭ̦Ҽ{�t��ơC�]�N�O�Ҽ{

��ؼs�q�n��C

��A�p�G �h

�]��

�ҥH�W�z�s�q�n�� p > 1 �ɵo��C �� p = 1 ��ɭԡA1/x ��Ͼɨ�ƬO ln(x)�AŪ�̥i�H�ۦ��A �� (0,1] ��s�q�n��]�o��C �ҥH�A�ڭ��o��סG

�A�̡A�p�G �h

�]��

�ҥH�W�z�s�q�n�� p < 1 �ɵo��C �� p = 1 ��ɭԡA1/x ��Ͼɨ�ƬO ln(x)�AŪ�̥i�H�ۦ��A ��

��s�q�n��]�o��C �ҥH�A�ڭ��o��סG

�H�W (1) ��M (2) ��ܮe��ɡA�ҥH��Ū�̤��n�I�w�C Ū�̩γ\�|ı�o (1) ��M (2) ��u��۬ۤϡv�A��S��C ��A��̭ǳz�S�F�@�ӳW�h�G

�O�@��ɽu�G�񥦡y�j�z��ơA�s�q�n��o��F �񥦡y�p�z��ơA�s�q�n��ġC

�W��y�ܫD�`��W�A��O��ӥi�H��Ū�̤@��²�K��p�ǫh�C �H�W�ҿת��u�j�v�M�u�p�v�A��O��Y�ƦӨ��A�Ҧp

��M��

�p�A��O�u�O�Y�ƪ��t�O�A�b��ƤW�A��٬O�P�@��šC �]��b (0,1] ��s�q�n��M�o��C �P�z�A

�]��O��

�u�j�v�C�H�W�һ��o�u�j�v�M�u�p�v�A��O��ơA �]�N�O (1) �M (2) �� p �C�Ӭ��|�o�˻��O�A �ݥH�U��i�ϥܴN��դF�C

x=0..1, p=0.8, p=1, p=1.2	x=1..15, p=0.8, p=1, p=1.2

�ڭ̬ݨ�A��⦱�u�O�u�ɽu�v�A�ӹ�� (1) ��M (2) ��A�|��Ī��u�`�b�u�U��v�C �p�G�ڭ��A�Ѿ��Ʀb (0,1] �M

�϶��u��Ƥj�p�v�P�u��Ƥj�p�v��Y�A�N��|ı�o (1) ��M (2) ��u�ۤϡv�F�A ��̤��S��u�ۤϡv�A�ϦӬO�@�Ӥ@�P��{�H�C

��F�קK�@�ǲӸ`�ް_��·СA�ڭ̱q��u��Q�D�t��ơA�]�N�O f(x) >= 0 ��ơC �p�G�J��ݭn�B�z�t�Ȩ�ƨ�ƪ��ɭԡA�q�`��䵴��ȧY�i�C �]��A�حz�C

�b�޳N�W�A��ڭ̭��@�� (a,b] �϶��W��u�L��j��ơv�s�q�n��A �n��ˤ~��_�w�� $x\to a^+$ ��ɭԡA $f(x)\to\infty$ ��t�פ� $\frac1{x-a}\to\infty$ ��C (�]��u��b��U��) �O�H�շQ�A�p�G�� x �a�� a ��ɭԡA �s�b�Y�� p < 1 �ϱo

$\begin{displaymath} f(x) \approx \frac1{(x-a)^p} \quad\hbox{where}\quad x\approx a \end{displaymath}$

�h�]��

$\begin{displaymath} \lim_{x\to a^+} \frac {x-a}{(x-a)^p} = \lim_{x\to a^+} (x-a)^{1-p} = 0 \quad (p<1)\end{displaymath}$

�ҥH�ڭ̤��סG�p�G

$\begin{displaymath} \lim_{x\to a^+} (x-a)f(x) = 0 \end{displaymath}$

�h

$\begin{displaymath} \int_a^b f(x) dx \end{displaymath}$

�O�i�n��F�_�h��i�n�C

�t�譱�A��ڭ̭��@�ӡu�L��϶��v�s�q�n��A �n��ˤ~��_�w�� $x\to\infty$ ��ɭԡA $f(x)\to 0$ ��t�פ� $\frac1x\to0$ �� (�]��u��b��U��) �O�H �շQ�A�p�G�� x �ܤj��ɭԡA�s�b�Y�� p > 1 �ϱo

$\begin{displaymath} f(x) \approx \frac1{x^p} \quad\hbox{where}\quad x\to\infty \end{displaymath}$

�h�]��

$\begin{displaymath} \lim_{x\to\infty} \frac{x}{x^p} = \lim_{x\to\infty} \frac1{x^{p-1}} = 0 \quad(p>1) \end{displaymath}$

�ҥH�ڭ̤��סG�p�G

$\begin{displaymath} \lim_{x\to\infty} xf(x) = 0 \end{displaymath}$

�h

$\begin{displaymath} \int_a^\infty f(x) dx \end{displaymath}$

�O�i�n��F�_�h��i�n�C

�Ҧp�A�Ҽ{

$\begin{displaymath} \int_0^1 \frac1{\root 3 \of{2x-x^2}} dx \eqno(3) \end{displaymath}$

�o�O�@�� (0,1] �϶��u�L��j��ơv�s�q�n��C�]��

$\begin{displaymath} \lim_{x\to 0^+} \frac{x}{\root 3\of{2x-x^2}} = \lim_{x\to 0^+} \frac{x^{2/3}}{\root 3\of{2-x}} = \frac0{\root 3\of 2} = 0 \end{displaymath}$

�ҥH�ڭ��_�w (3) �O�i�n��C

�A�Ҽ{�H�U�b�L��϶��s�q�n��G

$\begin{displaymath} \int_1^\infty \frac1{\sqrt{x^3+5}} dx \eqno(4) \end{displaymath}$

�]��

$\begin{displaymath} \lim_{x\to\infty} \frac x{\sqrt{x^3+5}}= \lim_{x\to\infty} \frac 1{\sqrt{x+\frac5{x^2}}}=0 \end{displaymath}$

�ҥH�ڭ��_�w (4) �O�i�n��C

�_�w�@�Ӽs�q�n��i�n�A�ä��N��D��סC �Ҧp�e�� (3) ��M (4) ��A��T�w�O�i�n��A �i�O�ڭ̫o�L�k�H�ثe�ҾǪ��n��ޥ��Ӻ�X��סC ��A�o�اP�_�z�צ��ΩO�H ��γ~�O�A�]��ƭȤ�k�g��q��{��A�`�O�|��X�@�ӵ��סA �p�G��_�w�i�n�A�h�ڭ̤��īH��ӵ��סC �ӧP�w�F�i�n��A�N��\�h�i�઺�ƭȤ�k�i��Q�ΡA �ΨӦ��p�s�q�n��ȡC �ܩ󨺨ǼƭȤ�k�A�h��O�@��L�n��ҵ{��Q�׽d��C

�Q�εL��s�q�n��U��ҩM�A�ڭ̥i�H�P�_�Y�ǵL�a�żƪ��ĩʡA �٥i�H��D�o�@�ӤW�ɡC�Ҧp�Ҽ{

$\begin{displaymath} \sum_{n=K}^\infty \frac1{n^3} \end{displaymath}$

�ڭ̯d��Ū�̦ۤv�h��ҡA�W�z�L�a�żƬO�H�U�s�q�n��D

$\begin{displaymath} \int_{K-1}^\infty \frac1{x^3}\,dx \end{displaymath}$

�H K, K+1, K+2, K+3, ... ��`�I�Ұ��X��U��ҩM�C�]��A

$\begin{displaymath} \sum_{n=K}^\infty \frac1{n^3} < \int_{K-1}^\infty \frac1{x^3}\,dx = \frac1{2(K-1)^2} \end{displaymath}$

�ҥH�ڭ̤��_�w��L�a�żƦ��ġA�٪��D�F��̤j�]��ܩ�W�L 1/(2*(K-1)^2)�C

��D

�а�
�а�

�O�_�i�n�H²�z�z�ѡC
�p�G

�а�

²�z�z�ѡC
�а�

�O�_�i�n�H²�z�z�ѡC
�p�G

�мg�X C �M g(x)�C
�а�

�O�_�i�n�H²�z�z�ѡC
�а�

�O�_�i�n�H²�z�z�ѡC
�а�

�O�_�i�n�H²�z�z�ѡC
�а�

�O�_�i�n�H²�z�z�ѡC
�а�

�O�_�i�n�H²�z�z�ѡC
�а�

�O��@�Ӱ϶��u�L��j��ơv�s�q�n��H�O�_�i�n�H²�z�z�ѡC
�аݡA��Ǽ� p �i�H�ϱo�H�U�s�q�n��i�n�H²�z�z�ѡC[HH]
�аݡA��Ǽ� p �i�H�ϱo�H�U�s�q�n��i�n�H²�z�z�ѡC[HH]

����ƪ��s�q�n��

���D

��ƪ��s�q�n��

��D