Example of an infinite sequence of irrational numbers converging to a rational number?

Are there any nice examples of infinite sequences of irrational numbers converging to rational numbers?

One idea I had was the sequence: $ 0.1001000010000001\cdots,0.1101000110000001\cdots,\cdots,0.1111000110000001\cdots,$ etc.

Where the first term in the sequence has ones in the place $i^2$ positions to the right of the decimal point. $(i=1,2,3,\dots)$ For the second term, we keep all the ones from the first term and add one in place of the first zero after the decimal point. We then add ones $i^3$ places after the decimal point. (The logical sum 1+1=1, i.e a number $i^2=j^3$ spaces after the decimal place has the value 1).

It is clear that this will converge to $1/9$, and I don't think the decimal expansion repeats at all.

Solution 1:

Pick any irrational number $\alpha$ you like, then consider the sequence $\{x_n=\alpha/n\}_{n=1}^\infty$. Then each term of the sequence $x_n$ is irrational and it converges to zero as $n$ tends to infinity.

Solution 2:

How about the sequence $\sqrt{2}/n$, converging to zero?

Or, if you want to see some pattern with the decimal expansion, how about $\frac{\sqrt{2}}{10^n}$ giving 0.141421..., 0.0141421..., 0.00141421... ?

Solution 3:

$\large\sqrt[n]{n}$ is irrational when $n>1$, and $\lim\limits_{n\to\infty}\large\sqrt[n]{n}=1$.

See also the related question Existence of irrationals in arbitrary intervals.