Accumulation Point

The idea sees substantial usage in an Advanced Calculus test. An Accumulation Point might also be called a Limit Point or a Cluster Point.

Explanation

An accumulation point fits the definition we build with the following scenario:

  • Let X be a collection of of points, with x \in X.
  • Let A be a collection of points where A is a subset of X.

A point ‘x’ in A is an accumulation point if for any positive value epsilon[\latex], you can build an open neighborhood around 'x'--with the neighborhood being all the points with a distance to 'x' that is less than [latex]epsilon[\latex], and there will be at least one point [latex]a \in A in that neighborhood.

Because the neighborhood is an open neighborhood, our point 'a' in the neighborhood doesn't have the option of sitting on the "fence" (with fence being the circle of points with a distance of epsilon[\latex] from the alleged accumulation point).</p> <p><strong>Examples</strong></p> <p>Every limit of a non-constant sequence is an accumulation point of the sequence.</p> <p>Consider:</p> [latex] \dfrac {1} {2}, \: \dfrac {2} {3}, \: \dfrac {3} {4}, \: \dfrac {4} {5}, \: \dfrac {5} {6}, ...

Our mathematical intuition tells us this is getting closer and closer to 1 but that we will never get there.

Consider a sequence defined by

\dfrac{1}{n}

where n is a real number. Your mathematical intuition tells you that as n gets larger this calculation gets closer and closer to zero.

The value zero is an Accumulation Point for this sequence, and here is why:

For any \epsilon where \epsilon > 0 there exists some n given n \in N where | \dfrac {1}{n} - 0 |< \epsilon.

This means we can get \dfrac {1}{n} - 0 as close as we want to zero, by making n larger.

Appendix N

Newton postulated that if we can get the difference between a calculation and a valued called "L" as small as we want by moving the variable closer and closer to a target called "c", then at the target, the difference between the calculation and "L" will be zero, and this means the value of the calculation is L. This is a fundamental of calculus and it is the "trickery" we can use to find L when legally the calculation gives the answer "division by zero error".