[saithound]

> I have no idea why you think the geometric series has anything to do with this -

IgorPartola is perfectly right to mention geometric series, you can easily use a geometric progression to construct a shape with infinite perimeter and finite area, e.g. by gluing together rectangles with height one and width decreasing in geometric progression. With a bit more thought you can also construct a smooth shape having this property.

[almostgotcaught]

> together rectangles with height one and width decreasing in geometric progression

The geometric series sums to 2 - your glued together rectangles will have perimeter 2*(1+2) and area 2*1.

[saithound]

> your glued together rectangles will have perimeter 2*(1+2)

No. You should think through that perimeter calculation one more time, preferably while drawing a picture.

Here's a hint: the perimeter of a rectangle is no less than its height; you can glue so that the perimeter of each rectangle contributes at least 1 to the perimeter of the union.