Sum of rational numbers

Solution 1:

Look at this.

So: $3.14159 \dots = 3 + \frac{1}{10} + \frac{4}{10^2} +\frac{1}{10^3} + \frac{5}{10^4}+ \frac{9}{10^5} + \dots$

The above expression is $\pi$.

Solution 2:

The problem is psychological: you think of the "infinite sum" of rational numbers as an obvious, intuitive concept, but it's not. It has a precise mathematical meaning, and that precise mathematical meaning only works if you allow the sums to be real numbers (which themselves have a precise mathematical meaning). The definitions which allow these "infinite sums" to make sense are much less trivial than someone who's never worked through them would think.

Solution 3:

Dear I am thinking about it because I am also a student of Mathematics.

We know that $e$ is an irrational number.

The value of $e$ is

$$= 1 + \frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \ldots$$

$$= 1 + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \ldots$$

$$= 2.7182 \ldots$$ (Irrational Number)

So sum of infinite rational numbers may be irrational.