In order to unlock the Sierpinski carpet problem, it is very important to define the area that is being taken away. There are two key elements of the area that is removed, the number of the area and the length of the area. If the area that is removed is found, it is easy to find the rest the area, which is basically just one minus the removed area. The last part of the problem requires looking back at the setting of the algorithm, which already gives the answer away.

In order to distinguish a convergent series from a divergent series, we have to know what makes a series converge.

First of all, a series is the sum of a sequence of numbers. Since I have already explained about convergence which is affected by boundary and monotone, we know that a convergent sequence will eventually get closer and closer to a certain number which we call L in the mathematical world. Similarly, with a series where the sum of the sequence is bounded and monotone and will get closer to L, it is defined as a convergent series. On the other hand, a divergent series is the sum of a sequence that does not get close to L.

An example of a convergent series is a person who is going on a one-mile walk. Let us assume that the person starts from zero position and zero time. The sequence is the distance that the person walks within a certain time. The series is the total distance that the person walks from the beginning. As we can see, the total distance is eventually getting closer and closer to one mile. Since this series is bounded between zero to one mile and is increasing and getting closer to one mile, it is convergent and converges to the L which equals to one in this case.

Similarly, an example of a divergent series is just like a person who does not know where to go. Since the person keeps going back and forth, the total distance is always changing and it never gets close to any point. This series diverges because it is not monotone and it does not reach to any L.

In conclusion, if a series or the sum of the sequence converges to L, then this series converges. Otherwise, it diverges.

In order to distinguish convergent sequences from divergent sequences, there are two key elements that we need to consider and those are boundary and monotone.

A convergent sequence is a list of elements that are within a boundary of two numbers. It cannot go beyond one number and it also cannot be below the other number. Another property of convergent sequence is monotone which means that it is always increasing or decreasing. Any sequence that is not convergent, diverges.

An example of a convergent sequence in real life is the age of a human being. As we know, a person’s age always increases by one every year. It cannot be higher than 200 (for now) and cannot be lower than zero. Therefore, the age of a human being is a convergent sequence.

Additionally, let us have an example for a divergent sequence. We can talk about the temperature in a day. As we know, the temperature reaches the highest point at noon and reaches the lowest point at night. In other words, it is increasing from night to noon and decreasing from noon to night which means it is not monotone. Therefore, we say the temperature in a day is a divergent sequence.

In conclusion, boundary and monotone are the key terms to distinguish convergent sequences from divergent sequences.