# Is there such a thing in math the inverse of a sequence?

Such as can I construct a sequence by reversing the order of the approximating sequence of $$\frac{1}{3}$$? So such inverse would look like $$\left\{….,0.333333333,0.3333333,0.333333,...0.3\right\}$$.

I had this question when I was constructing a sequence that is bounded between 0 and 1/2 and not convergent to 0 for a homework question.

• You can't, imagine if the first element has $n$ three's but in the original sequence the element with $n+1$ three's come after the one with $n$ three's hence in the new sequence it should be before the first element, a contradiction. Commented Mar 8, 2019 at 1:50
• And no, there is a difference between stupid and bad questions, in your case the votes have matchedvso your question divides opinion. Giving enough context helps when you are asking a "stupid" question, i.e. mentioning that you heard it in class or in conversation with a friend or online on some webpage helps. Also mention what was the broad topic if the conversation, so that others will be able to address concerns relating to that in the answer as well. Commented Mar 8, 2019 at 3:44
• This is a good question. In several cases, a sequence makes sense for negative values of the index. For example, the Fibonacci numbers $\{0,1,1,2,3,5,\dots\}$ can be exteneded to negative indices as $\{\dots,5,-3,2,-1,1\}$. I encounter this kind of thing in the OEIS many time. Commented Mar 8, 2019 at 5:45

A sequence is ultimately a map whose domain is $$\mathbb N$$, but the object that you describe (a) has no first term, and (b) terminates.
If you want, I suppose you could define an object whose indices run from $$-\infty$$ to $$0$$, but that's not appreciably different from looking at the original sequence while standing on your head.