Continued Fractions

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Continued Fractions

This 'infinite fraction' can be considered as a sequence of terms, tn:

A general formula for tn+1 in terms of tn can now be determined. It can be seen that tn+1 is 1added to 1 divided by the previous term.

i.e.

Decimal equivalents of each term can be computed. Here are the values for the first ten terms (correct to 5d.p as we can then see how the numbers differ):

t1=1

t2= 1.50000

t3= 1.66667

t4= 1.60000

t5= 1.62500

t6= 1.61538

t7= 1.61905

t8= 1.61765

t9= 1.61818

t10= 1.61798

From looking at the graph we can see that for the first few terms the values fluctuate, but eventually the values fluctuate less and become very close together. I.e. the values become closer together as the value of n increases.

We can then conclude that as n increases tntn+1. From this, we can now deduce a formula for tn+1 in terms of tn:

We can also conclude that if tntn+1, then tn-tn+1 will equal to zero.

Any term (the nth term) can be determined using the above