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{\large ���wI���K�@��P�Q��@2007�N12��11���z�z} 

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�S�� ���n����CTA �O�p �~
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{\small

���K����
{\tt http://www.ms.u-tokyo.ac.jp/$\sim$hirachi/courses/sugaku-I-2007/}
����_�E�����[�h�ł��܂��D�u�`�������ڂ��Ă��܂��D
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{\bf ���.}
���̓����������F
$$
\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}=
\int_0^1\int_0^1\frac{dxdy}{1-x^2y^2}
=\frac{\pi^2}{8}
$$
���K�ŏЉ��̂�
D. Zwillinger, Handbook of integration,  Jones and Bartlett Publ., 1992, pp. 37--38.
�ɂ�鋰�낵���I���Ȍv�Z�G���ʂ͎v�����Ȃ��B
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{\small
�ȉ��̖����ł���͈͂ŉ���, 12��18��13���܂łɃA�h�~�j�X�g���[�V�������̃��|�[�g��o�{�b�N�X�ɒ�o���邱��. 
}

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\prob �ȉ��̏d�ϕ����v�Z����

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(a) 
$\displaystyle{\iint_D(x^2+y^2)dxdy}$, $D$��$(1,1)$�𒆐S�Ƃ邷���a1�̉~��
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(b)
$\displaystyle{\lim_{K\to\infty}\iint_{\{x^2+y^2\le K\}}\frac{xy}{(1+x^2+y^2)^3}dxdy}$

\prob 
$u=x^2-y^2$, $v=2xy$�ŗ^������$xy$���ʂ���$uv$���ʂւ̎ʑ����l����B

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(a) $xy$���ʂłR�̒���$y=x$, $y=1$, $x=0$�ň͂܂��̈��$G$�Ƃ���B��̎ʑ��ɂ��$G$�̑������߂�B

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(b)  
$\displaystyle{\iint_G\frac{x^4-y^4+2xy(x^2+y^2)}{(1+x^2-y^2)^{1/2}}dxdy}$
�����߂�B


\prob ����$\{a_n\}$��$a_n>0$�𖞂����Ƃ���B$\lim_{n\to\infty}a_{n+1}/a_n>1$�ł����
$\sum_{n=1}^\infty a_n$�͔��U���邱�Ƃ������B(�ȒP������Ǝv���l�͋Ɍ�����Ɍ��ɂ����Ď����B)

\prob ���̋����͎������邩�H
$$
\sum_{n=1}^\infty\frac{1}{1+2^n},\qquad
\sum_{n=1}^\infty\frac{1}{1+\sqrt{n}},\qquad
\sum_{n=1}^\infty\frac{\sqrt{n}}{1+n^2}.
$$


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