计算:
1
1
?
2
+
1
2
?
3
+
1
3
?
４
＋ ．．．＋
1
98
?
99
+
1
99
?
100
\frac{1}{1 * 2} + \frac{1}{2 * 3}+ \frac{1}{3 *４}＋ ．．．＋\frac{1}{98 * 99} + \frac{1}{99 * 100}
1?21?+2?31?+3?４1?＋ ．．．＋98?991?+99?1001?
1
1
?
2
+
1
2
?
3
+
1
3
?
4
＋ ．．．＋
1
98
?
99
+
1
99
?
100
\frac{1}{1 * 2} + \frac{1}{2 * 3}+ \frac{1}{3 *4}＋ ．．．＋\frac{1}{98 * 99} + \frac{1}{99 * 100}
1?21?+2?31?+3?41?＋ ．．．＋98?991?+99?1001?
=
(
1
1
?
1
2
)
+
(
1
2
?
1
3
)
+
(
1
3
?
1
4
)
+
.
.
.
.
.
+
(
1
98
?
1
99
)
+
(
1
99
?
1
100
)
= (\frac{1}{1} - \frac{1}{2}) +(\frac{1}{2} - \frac{1}{3}) +(\frac{1}{3} - \frac{1}{4}) +.....+(\frac{1}{98} - \frac{1}{99})+(\frac{1}{99} - \frac{1}{100})
=(11??21?)+(21??31?)+(31??41?)+.....+(981??991?)+(991??1001?)
=
1
?
1
100
=1 - \frac{1}{100}
=1?1001?
=
99
100
=\frac{99}{100}
=10099?
把一个分数拆成两个分数相减的形式，就是裂项法，可以根据裂项工式:
1
a
?
b
=
(
1
a
?
1
b
)
?
1
b
?
a
\frac{1}{a * b} = (\frac{1}{a} - \frac{1}{b}) * \frac{1}{b - a}
a?b1?=(a1??b1?)?b?a1?把一个数裂项为两个分数求差，然后前后抵消求和。裂项法是分解与组合思想在数列求和中的具体应用，通常用于代数，分数，有时候也用于整数数列求和。
=
1
2
(
1
1
?
2
?
1
2
?
3
+
1
2
?
3
?
1
3
?
4
+
1
3
?
4
?
1
4
?
5
+
.
.
.
.
.
+
1
97
?
98
?
1
98
?
99
+
1
98
?
99
?
1
99
?
100
)
=\frac{1}{2}(\frac{1}{1*2} - \frac{1}{2*3} +\frac{1}{2*3} - \frac{1}{3*4} +\frac{1}{3*4} - \frac{1}{4*5} +.....+\frac{1}{97*98} - \frac{1}{98*99}+\frac{1}{98*99} - \frac{1}{99*100})
=21?(1?21??2?31?+2?31??3?41?+3?41??4?51?+.....+97?981??98?991?+98?991??99?1001?)
=
1
2
(
1
1
?
2
?
1
99
?
100
)
=\frac{1}{2}(\frac{1}{1*2} - \frac{1}{99*100})
=21?(1?21??99?1001?)
=
4949
19800
=\frac{4949}{19800}
=198004949?