4 ms·
The sum, 0.9 * Σ (0.1)^n from n=0 to n=N-1, is 0.999... as N goes to ∞ (write out the terms). On the other hand, for any finite value of N, the sum [1] is equa
by chestervonwinch 10y ago
The sum, 0.9 * Σ (0.1)^n from n=0 to n=N-1, is 0.999... as N goes to ∞ (write out the terms).
On the other hand, for any finite value of N, the sum [1] is equal to 0.9 * (1-(0.1)^N) / (1-0.1) = 1 - (0.1)^N. This value goes to 1 as N goes to ∞.
More rigorously, for any positive value, ε, there is a value N, such that the value of the finite sum is within ε of 1.
[1]: https://en.wikipedia.org/wiki/Geometric_series#Formula https://en.wikipedia.org/wiki/Geometric_series#Formula