Question

Binomial Expansion
prove that:

1/1.3 +1/2.5 +1/3.7+1/4.9+....=2(1-ln2)

Answer 1

limn→∞[12(3−11.3)+12(5−33.5)+...+12((2n+3)−(2n+1)(2n+1)(2n+3))] lim n → ∞ [ 1 2 ( 3 - 1 1.3 ) + 1 2 ( 5 - 3 3.5 ) + ... + 1 2 ( ( 2 n + 3 ) - ( 2 n + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ) ] limh→∞[12(1−13+(13−15)+(15−17)+...+(12n+1−12n+3)] lim h → ∞ [ 1 2 ( 1 - 1 3 + ( 1 3 - 1 5 ) + ( 1 5 - 1 7 ) + ... + ( 1 2 n + 1 - 1 2 n + 3 ) ] limh→∞12(1−12n+3) lim h → ∞ 1 2 ( 1 - 1 2 n + 3 ) limh→∞12(2n+22n+3) lim h → ∞ 1 2 ( 2 n + 2 2 n + 3 ) limh→∞12(2n(1+1n)h(2+3n)) lim h → ∞ 1 2 ( 2 n ( 1 + 1 n ) h ( 2 + 3 n ) ) 122(1+0)2+0 1 2 2 ( 1 + 0 ) 2 + 0 12 1 2 .