3 ms·
The sequence doesn't end earlier, but it does start later by one element, so they are not exactly same sequences. One infinite sequence has one more element tha
by drran 6y ago
The sequence doesn't end earlier, but it does start later by one element, so they are not exactly same sequences. One infinite sequence has one more element than another infinite sequence, so difference is 1/inf.
- nybble41 6y agoNo, they start at the same element (they're both 0.999… and thus start with 9/10) and have the same (infinite) "number" of elements. If you lined them up digit by digit there is never a point where one digit is a 9 and the other is a 0.
- nybble41 6y agoIncidentally, this confusion is part of the reason why actual infinite sequences are written with a trailing ellipsis (9/10 + 9/10² + 9/10³ + …) and not a final term involving infinity. There is no final term—not even 9÷10^∞. So the correct way to write the separation of the leading term in your formula is: x = 9÷10 + 9÷10² + 9÷10³ + … x = 9÷10 + (9÷10 + 9÷10² + …) ÷ 10 Note that the number of elements in the sequence is the same no matter how many leading terms you write, so long as the pattern doesn't change. The notation { 1, 3, 5, 7, 9, … } and { 1, 3, 5, … } both refer to exactly the same set; the first notation is merely a bit more verbose. Similarly, the parenthesized portion of the second formula above is exactly equal to x, despite being written with two explicit leading terms rather than three.