Monday Math 37

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 Fn.