00:01
To find the sum of the telescoping series, we first want to rewrite 1 over the product of k plus 6 and k plus 7 as a sum of partial fractions.
00:12
Now, 1 over k plus 6 times k plus 7, that's equal to the sum of partial fractions, which denominators are k plus 6 and k plus 7.
00:28
So since both are linear, then their numerators are constant.
00:33
Let's call them a and b.
00:35
If i multiply this, both sides, by the lcd, k plus 6 times k plus 7, we have one that's equal to a times k plus 7 plus b times k plus 6.
00:54
If k plus 7 equals 0, that means k equals negative 7, then we have 1 equal to 8 times 0 plus b times negative 6, that's 1 equal to negative b, or that means b is equal to negative 1.
01:22
If k plus 6 is instead 0, then k equals negative 6 and 1 is equal to a times negative 6 plus 7 plus b times 0, we get 1 equal to a.
01:45
So 1 over k plus 6 times k plus 7, which is a over k plus 6 plus 6 plus 6 plus 6 plus 6 plus 6 plus 6 plus b over k plus 7 is equal to 1 over k plus 6 minus 1 over k plus 7...