## a and 1/a

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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:

$\frac{1}{a}=x+x^2+x^3+...$

The first term and ratio of this geometric series both equal $x$ and so the series sums to $\frac{x}{1-x}$.  But we said this series was equal to  $\frac{1}{a}$ so it follows that

$a=\frac{1-x}{x}=\frac{(1-x)}{1-(1-x)}$, which is the sum of the geometric series whose first term and ratio both equal $(1-x)$.  In other words,

$a = (1-x)+(1-x)^2+(1-x)^3...$

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

[As a further aside, if $a$ is an integer then it turns out that $x=\frac{1}{a+1}$.  For example, $\frac{1}{4}=(\frac{1}{5})+(\frac{1}{5})^2+(\frac{1}{5})^3...$ and therefore $4=(\frac{4}{5})+(\frac{4}{5})^2+(\frac{4}{5})^3...$.  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.