Calculus Tutorials - Sequences
A sequence corresponds to infinite array or list of number of the form
where are real numbers. For example, the sequence
is represented by the list
because those are the values the expression takes when takes the values 1, 2, 3, ...etc.
Convergence of sequences
One concept that is typically hard to grasp is the convergence of a sequence. The idea is very trivial though: A sequence
converges to a value if the values of the sequence get closer and closer to (in fact they get as close as we want) as approaches to infinity.
For example: The sequence is such that
because the value of becomes "as close to zero as we want" as approaches to infinity.
Formal defintion of convergence: The sequence as , or otherwise said
For all , there exist such that
This is saying that no matter how close you want the sequence from , there is always a point in the sequence such that
all the points further than that, are close enough to . In other words the convergence of a sequence doesn't state that
some number of the sequence get close enough to the limit , but instead, it indicates that if we go far enough into the sequence,
all the values of if will be close enough.
Algebra of Limits
Operating with limits is not as complicated once we know some them. In fact, there are simple rules that allow to compute
more complicated limits based on simpler ones. These rules are shown below:
If and then we have:
(where property (3) holds as long as .)
Example: The limit
is computed by first multiplying both numerator and denominator by , which means