Editing Simple continued fraction (section)

=== Interval for a convergent ===
A rational number, which can be expressed as finite continued fraction in two ways,
:{{math|''z'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}}, 1] {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}} + 1] {{=}} {{sfrac| ''p''{{sub|''k''}}|''q''{{sub|''k''}}}}}}
will be one of the convergents for the continued fraction expansion of a number, if and only if the number is strictly between (see [https://math.stackexchange.com/a/4438961 this proof])
:{{math|''x'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}}, 2] {{=}} {{sfrac| 2''p''{{sub|''k''}} - ''p''{{sub|''k-1''}}|2''q''{{sub|''k''}} - ''q''{{sub|''k-1''}}}}}} and
:{{math|''y'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}} + 2] {{=}} {{sfrac| ''p''{{sub|''k''}} + ''p''{{sub|''k-1''}}|''q''{{sub|''k''}} + ''q''{{sub|''k-1''}}}}}}
The numbers {{mvar|x}} and {{mvar|y}} are formed by incrementing the last coefficient in the two representations for {{mvar|z}}. It is the case that {{math|''x'' < ''y''}} when {{mvar|k}} is even, and {{math|''x'' > ''y''}} when {{mvar|k}} is odd.

For example, the number {{sfrac|355|113}} ([[Milü|Zu's fraction]]) has the continued fraction representations
:{{sfrac|355|113}} = [3; 7, 15, 1] = [3; 7, 16]
and thus {{sfrac|355|113}} is a convergent of any number strictly between
:{| cellpadding="2" cellspacing="0"
| align="right" | {{math|[3; 7, 15, 2]}} ||{{=}}|| {{math|{{sfrac|688|219}} ≈ 3.1415525}}
|-
| align="right" | {{math|[3; 7, 17]}} ||{{=}}|| {{math|{{sfrac|377|120}} ≈ 3.1416667}}
|}