Yahoo Answers is shutting down on 4 May 2021 (Eastern Time) and the Yahoo Answers website is now in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Why is Ln(2) not 0 rather than 0.693..?
Ln(2) i.e the natural logarithm of 2 to base e is 0.693....
It can be calculated as Ln(2) = 1 - 1/2 + 1/3 - 1/4 + 1/5 ...
However this can be rearranged as
1 - 1/2(1 + 1/2 + 1/4 + 1/8 + 1/16 + ....) which equals 0 [ This is the same as 1 - 1/2 - 1/4 - 1/8 - 1/16 - 1/32 ...] +
1/3 - 1/6(1 + 1/2 + 1/4 + 1/8 + 1/16 + ...) which equals 0 +
1/5 - 1/10(1 + 1/2 + 1/4 + 1/8 + 1/16 +...) which equals 0 + ...
etc - in short for every odd fraction you can group a unique set of fractions to cancel it out
Which makes Ln(2) = 0!
2 Answers
- δοτζοLv 71 decade agoFavourite answer
You can't rearrange the terms of a conditionally convergent series.
1 - (1/2) + (1/3) - ... = ∑(-1)ⁿ / n
The absolute value ∑1/n isn't convergent, but ∑(-1)ⁿ/n is. It is therefore conditionally convergent and it's terms cannot be rearranged.
- ?Lv 44 years ago
i could incredibly be length sixteen. the way I see it, i could incredibly be chuffed with myself the way i grew to become into than obsessed approximately my seems and hectic that i do no longer yet look adequate like Skeletor to bypass as particularly. of path, once you're clearly a length 0 i do no longer see something incorrect with it, yet quite often, length 0 is a ludicrous objective and intensely unnatural.
