, 2, 4, 8 + 2 + 4 + 8 級數, 2, 4, 8,... + 2 + 4 + 8 +... 無窮級數 r = r < 2 + 2 + 4 + 8 +... = 2 = 2 2 2 + 2 + 4 + 8 +... + ( ) + + ( ) + +...
( + ( )) + ( + ( )) +... 0 + (( ) + ) + (( ) + ) +... {a n } = a, a 2, a 3,..., a n,... S n = a + a 2 + a 3 +... + a n = n k= a k S n partial sum n a n n= lim S n = lim n a k = S k= 2
n S n = + ( ) + +... + = n S n = + ( ) + +... + ( ) = 0 lim S n S n 0 r = n a + a r +... + a r n = a ( r n ) r S n a a ( r n ), r < S = lim S n = lim r r, r < a r S n = a ( r n ) lim S n r 2 n n! 3
A B a n b n A B c m. ca n ca 2. (a n + b n ) A + B 3. a n n=m a n + cb n c a n b n a 0 a m 4
( ) 2 n ( ) 6 n + 3 7 ( ) 2 n ( ) 6 n 3 7 2 3 + 6 7 = 3 + 7 = 0 2 n + 3 n 6 n 2 n + 3 n 6 n = 2n 6 n + 3n 6 n = 3 n + 2 n 3 n + 2 n 3 + 2 = 3 2 + 2 = 7 2 5
3 n+ π n 3 3 3 n ( 3 n π π) = 3 n 3 π e 3n ( e 3 ) n e 3 6
a n b n c n c m. ca n 2. (a n + c n ) 3. (a n + b n ) 4. a n n=m 0 a n = 3 n n b n = n a n +b n = 3 n a n = ( ) n b n = ( ) n+ a n + b n = 0 a n 0 L L 7
Theorem.. a n n= lim a n = 0 an S S n = a +... + a n S n = a +... + a n a n S n S n lim a n = lim (S n S n ) = lim S n lim S n = S S = 0 lim a n = 0 + 2 + 3 + 4 + 5 +... a n n () + ( 2 ) + ( 3 + 4 ) + ( 5 +... + 8 ) +... 2 n n= () + ( 2 ) + ( 3 + 4 ) + ( 5 +... + 8 ) +... () + ( 2 ) + ( 4 + 4 ) + ( 8 +... + 8 ) +... 8
() + ( 2 ) + ( 4 + 4 ) + ( 8 +... + 8 ) +... = () + ( 2 ) + (2 4 ) + (4 8 ) +... = + 2 + 2 + 2 +... 2 + 2 + 3 +... + n +... n 2 3n + 6 2n 2 + 8n 3 n2 3n + 6 2n 2 + 8n 3 lim n 2 3n + 6 2n 2 + 8n 3 = 2 0 9
sin(n) lim sin(n) n= n(n + ) n(n + ) = n n + n S n = k(k + ) k= n = k k + k= ( ) ( ) ( ) = + 2 2 + 3 3 +... + 4 = n + 0 ( n ) n +
n= n(n + ) = lim S n = lim n + = telescoping series Oxford Advanced Learner's Dictionary telescope. to become shorter, or make something shorter, by sliding sections inside one another. 2. to reduce something so that it happens in less time. 嵌疊級數 n + 3 n + 2 n S n = k + 3 k + 2 k=0 = ( 3 2 ) + ( 4 3 ) +... + ( n + 3 n + 2 ) = n + 3 2
n= n + 3 n + 2 = lim S n = lim n + 3 2 = arctan(n + ) arctan(n) S n = n arctan(k + ) arctan(k) k=0 = (arctan() arctan(0)) +... + (arctan(n + ) arctan(n)) = arctan(n + ) arctan(0) = arctan(n + ) arctan(n + ) arctan(n) = lim S n = lim arctan(n + ) = π 2 2