How do we find the decimal (or base) expansion of a real number upto first n digits, given a sequence that converges to it?

Question

Suppose we have a sequence ${x_n}$ that converges to some limit, say $\pi$. How do we compute out the decimal expansion of $\pi$ from that sequence? How could the epsilon definition help us here? What are some considerations to have the decimal expansion of the number is not unique?

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On existence and Uniqueness of decimal expansion

Answer 1

For the case of real numbers with unique expansions:

It seems so this sort of thing is very easy to answer once we have a real between the epsilon and the sequence index.

Let us recall the definition, if for any $\epsilon >0$, we have an $N$ such that for all $n>N$, we have that $|x_n -L| < \epsilon$.

If we were to put that epsilon is $\frac{1}{10^1}$, then it would mean that would give us an $N$ for which, the sequence terms agree to the limit upto ones place.

And, if we choose it to be $\frac{1}{10^2}$ the terms of the sequence after the index $N$ would be right upto first decimal digit

Similarly, if we choose epsilon to be $\frac{1}{10^{i+1}}$ and find the corresponding $N$ from that we would find all the the numbers which agree to our limit upto $i$ decimal places.

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What's interesting is, say we were in another base, say 2 maybe, then we could use similar procedure by considering $\frac{1}{2^i}$

Note: A given real number may have many decimal expansions, but for a given sequence that converges to it, it sho