Thanks Dr. Gowers’. These are invaluable insights into basics. Thanks for giving so much of your time.
In the Feb 23 2018 blog problem, we posed the following question:
Sum the following infinite series:
The sum can be written as:
, where .
Thus, . This is the answer.
If you think deeper, this needs some discussion about rearrangements of infinite series also. For the time, we consider it outside our scope.
Let be a smallest value of the function . Prove that when .
From this, we see that for and . Consequently, attains its maximum value in the interval . On this interval
So, . But,
As , the first term on the right hand side tends to the limit . In the second term, the factor
of the numerator tends to zero because
Reference: Nordic Mathematical Contest, 1987-2009.
Reference: Nordic Mathematical Contest 1987, R. Todev:
Let f be a function, defined for natural numbers, that is strictly increasing, such that values of the function are also natural numbers and which satisfies the conditions and for all natural numbers m and n. Define the smallest possible value of a.
Since, is a function satisfying the conditions of the problem, the smallest possible a is at most 4. Assume that . It is easy to prove by induction that for all . So, taking into account that f is strictly increasing, we get
as well as .
So, we arrive at . But, this is not possible, since is an integer. So, .