Alicia wants to buy a carpet to cover her whole living room, except for the tiled floor. The tiled floor is 6 1/2 ft. by 3 1/4 f
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An alternating series 
converges if 
is monotonic and 
as 
. Here 
.
Let
. Then 
, which is positive for all 
, so 
is monotonically increasing for . This would mean 
must be a monotonically decreasing sequence over the same interval, and so must 
.
Because is monotonically increasing, but will still always be positive, it follows that as .
So, converges.
Let
Because
So,