https://gowers.wordpress.com/2008/07/30/recognising-countable-sets/

Thanks Dr. Gowers’. These are invaluable insights into basics. Thanks for giving so much of your time.

https://gowers.wordpress.com/2008/07/30/recognising-countable-sets/

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:

.

**Solution:**

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.*

*Cheers,*

Nalin Pithwa.

**Problem:**

Let be a smallest value of the function . Prove that when .

**Proof:**

For ,

.

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

.

So,

auf wiedersehen,

Nalin Pithwa.

**Reference: Nordic Mathematical Contest, 1987-2009.**

**Reference: Nordic Mathematical Contest 1987, R. Todev:**

**Question:**

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.

**Solution:**

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, .

Cheers,

Nalin Pithwa.