a and 1/a

In the last Carnival of Mathematics, two of the number facts were:

• 1/49 is the sum of the series 0.02+0.0004+0.000008+…
• 49 is the sum  the series 0.98+0.98²+0.98³+…

(Incidentally, I originally tried to put parentheses around the 0.98, but having an 8 and ) next to each other made the end format as 8) and that looked pretty funny.)

It turns out that it’s not a coincidence that the ratios in the two geometric series (0.02 and 0.98 ) add to 1.  As proof, suppose that:

The first term and ratio of this geometric series both equal and so the series sums to .  But we said this series was equal to  so it follows that

, which is the sum of the geometric series whose first term and ratio both equal .  In other words,

Not the most exciting fact in the world, but still intriguing.

[As a further aside, if is an integer then it turns out that .  For example, and therefore .  I’m not sure if this makes the series more or less interesting, so I’ll pretend the answer is more.]

Blame Credit for this post actually goes to TwoPi, who first came up with the sums for 49 and 1/49 and who noticed the pattern of adding to 1.  Credit for the photo goes to Arjan Dice; it’s published here on Wikipedia.

Advertisements

Tags: Geometric Series