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.