Final answer:
The sum of the series ∑ (1/n+2 - 1/n+1) is (1/3) - (1/(n+1)).
Step-by-step explanation:
To find the sum of the series ∑ (1/n+2 - 1/n+1), we can rewrite the terms as (1/(n+2)) - (1/(n+1)).
Now, let's expand the series by writing out the terms for n = 1, 2, 3, and so on:
(1/3 - 1/2) + (1/4 - 1/3) + (1/5 - 1/4) + ...
We can see that the numerator of each term is 1, and the denominators form an arithmetic progression. Using this information, we can simplify the series to (1/3) - (1/2) + (1/4) - (1/3) + (1/5) - (1/4) + ...
Notice that the negative terms and positive terms cancel each other out, leaving us with only the first and last terms. So, the sum of this series is (1/3) - (1/(n+1)).