When written as a decimal, the fraction 1/89 consists of a repeating decimal; specifically with a 44-digit repeat:

1/89=0.01123595505617977528089887640449438202247191…

Note the first few digits after the decimal point: 0, 1, 1, 2, 3, 5. You should recognise from last week the start of the Fibonacci sequence. Examing this, we see:

Or, using our convention last week, it looks like the series

is equal to 1/89. Does the sum actually hold? What if we try it in bases other than base 10?

In general, with base *b* (*b*≥2, of course), our sum in this fashion is:

Now, recall that last week we found the formula

,

where is the golden ratio.

This means our sum is:

.

Note that both sums in the result above are convergent geometric series, and as , we see that

,

and for *b*=10, , so the original infinite sum is 1/89. We see that for binary (*b*=2), we have the sum

.

Note that the zeroes of the quadratic polynomial are and , which are the terms raised to the power of *n* in our formula for *F*_{n}.

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Tags: Base, Binary, Decimal, Fibonacci Sequence, Geometric Series, Math, Monday Math, Repeating Decimal

This entry was posted on September 15, 2008 at 12:01 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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