I had so much fun deriving that series which converges to pi the other day that I thought it’d be fun to occasionally whip up such series for other irrational numbers. Today I’ll do .
There are many different series that would do the job, the one I’ll derive is found by expanding around as a power series.
Indeed, for satsfying we have
This series is guaranteed to converge for all , and to diverge for all and . The number 2 itself sits right on the boundary of the circle of convergence, so although we would like to say
we first need to show that the series is convergent. Consider the sequence . This sequence is monotonically decreasing, and we will show that for all . Firstly,
We now assume that for some integer and show that . This is true, since
where the last inequality holds for . Thus implies , and so by induction for all .
To show convergence of our series, we note we have shown enough to invoke the alternating series test. So it is true that
converges, and by continuity it must equal . Truncating the series at n=10,000,000 gives
which is correct up 11 decimal places.
Trigger Warning: This post takes the position that Santa Claus doesn’t exist.
I heard this one from a friend of a friend, it’s a good one. It goes like this:
Consider the logical connective associated with implication, the material conditional as it is sometimes called. It basically captures the intuition behind the “if blah is true then bleep is true”. So if and are two truth values (either true or false) and we write the material conditional as “if then “, then we can write out the four possible outputs:
- “if false then false” is true (both are false, implication still makes sense).
- “if false then true” is true (remember implies , but can be true whether or not is).
- “if true then false” is false ( implies , so this combination is not allowed).
- “if true then true” is true ( forces to be true).
The material conditional evaluates to either true or false, depending on whether the truth values are consistent with the implication. Now we can consider to represent the proposition “God exists”. Let stand for “Santa Claus exists”. We can now ask ourselves, is true? Well, clearly is false. What about the implication “if then “? I think everyone would agree that the existence of God in no way implies the existence of Santa, so we can say “if God exists then Santa exists” is false. We now check out the above list and observe that case 3 uniquely matches our situation; the whole implication is false and is false. From this we conclude that must be true, and therefore God exists.