Your base case is incorrectly worked out to start with.
The sum from i=1 to i=2 is 1/3+1/7=10/21
which is <1 so the base case is ok.
You have the P(k) proposition as
Then your P(k+1) proposition is
Then you try to prove that IF P(k) is true THEN P(k+1) will be true.
Do this by using the fact that if Pk) is true,
P(k+1) LHS will be less than or equal to
and hopefully this will be
Therefore we can ask if