Question

Give a sequence of rational numbers that converges to √5 (i.e. converges to L where L^2=5). No proof needed.

Answer 1

We have, = 2.23606797749978969640...

Now make a sequence as

2 , 2.2 , 2.23 , 2.236 , 2.2360 , 2.23606 , 2.236067 , . . .

That is, n-th term of the sequence is the decimal expansion of upto n-th term. In this sequence each term is a decimal number with finite decimal expansion. And we know that, any decimal number with finite decimal expansion is rational. Hence, each term of this sequence is rational.

Hence, this is a sequence of rational numbers that converges to .

The formal definition of this sequence is,

= , where is the n-th term of this sequence.

Here, for any real number x, denotes the greatest integer less than or equal to x.

So, {} is a sequence of rational numbers that converges to .