Today, I studied sequences, which form the very foundation of topics I will learn later such as limits, differentiation, and integration.
Through this post, I want to organize what I learned in a way that is easy to understand and enjoyable to read.
What Is a Sequence?
In one simple sentence, a sequence can be defined as “a list of numbers arranged in order.” The numbers themselves can be of any type.
Natural numbers, fractions, decimals, and all other kinds of numbers can be included.
In other words, regardless of the type of numbers, if numbers are arranged with a clear order, they can be called a sequence.
What Does “Order” Mean in a Sequence?
The word “order” in a sequence does not mean numerical order (from smaller numbers to larger ones).
For example, the sequence 3, -1, 7, 0, 12, … may look random in terms of size. However, since the first term is 3, the second is -1, and the third is 7, the position of each number is clearly defined.
For this reason, it is still considered a sequence. In a sequence, what matters is not whether a number is large or small, but which position the number occupies.
Valid Examples of Sequences
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1, 2, 3, 4, 5, …
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0.1, 0.01, 0.001, 0.0001, …
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3, -1, 7, 0, 12, …
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√2, 1.4, 1.41, 1.414, …
All of these examples share one common feature, the numbers are arranged in a specific order. Whether or not there is a clear pattern,
and whether the values increase or decrease, does not matter.
As long as the order is defined, the list is a sequence.
Examples That Are Not Sequences
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{1, 2, 3} (a set)
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“any numbers”
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“1 and 5 and 3” (when the order is unclear)
In these cases, numbers do exist, but since there is no clearly defined order, they cannot be called sequences.
What Is Convergence?
Now that we understand sequences, let’s look at what convergence means.
Convergence refers to a situation where, as a sequence continues, its values get closer and closer to a specific number.
For example, consider the sequence 0.1, 0.01, 0.001, 0.0001, 0.00001, …
As the sequence goes on, the values become smaller and smaller, and we can say that they are “approaching 0.”
This idea of “approaching” is exactly what convergence means.
Therefore, we say that this sequence converges to 0.
An important point here is that convergence does not mean
the sequence must actually reach that number.
It is not that the value fails to reach the number, but rather that convergence does not care about reaching it at all. In short, convergence refers to the phenomenon of getting closer to a specific value.
Convergence is about approach, not arrival.
What Is Divergence?
If convergence means “getting closer to a specific value,” then divergence can be thought of as the opposite concept. In convergence, there exists a specific reference value that the sequence approaches.
In divergence, however, no such reference value exists. In other words, divergence means that even as the sequence continues, there is no specific number that it can be said to approach.
If the behavior of a sequence cannot be described by a single number,
the sequence is said to diverge.
Common Types of Divergence
Divergence can be classified into several common cases.
Values Increase Without Bound
For example, the sequence 1, 2, 3, 4, 5, … continues to grow larger and larger and never stays near any specific number. In this case, the sequence is said to diverge to positive infinity.
Values Decrease Without Bound
The sequence -1, -2, -3, -4, -5, … continues to decrease indefinitely and likewise does not approach any specific number. In this case, the sequence is said to diverge to negative infinity.
Values Oscillate Back and Forth
The sequence 1, -1, 1, -1, 1, -1, … does not approach any specific number, nor does it move consistently in one direction. When values move back and forth in this way and never settle down, the sequence is said to diverge by oscillation.
The Difference Between Convergence and Divergence
To summarize…
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Convergence
→ There exists a specific reference value
that the sequence approaches.
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Divergence
→ There is no reference value
that the sequence can be said to approach.
Therefore, any sequence that does not converge is considered to diverge.
Divergence does not mean that values simply keep appearing,
but that the behavior of the sequence cannot be described by a single number.
What Is a Limit?
After learning about sequences, convergence, and divergence, a natural question arises. “When a sequence is said to converge to a certain value, what concept represents that ‘certain value’?”
The answer to that question is the limit. In simple terms, a limit means “the target value that something gets closer to as it continues.”
In other words, a limit is not about whether a value is actually reached, but about where it is approaching.
Limits in Sequences
Consider the sequence 0.1, 0.01, 0.001, 0.0001, … As the sequence continues, the values become smaller and smaller, and we can say that they get closer to 0.
That is why we say, “this sequence converges to 0.” Here, the ‘certain value’—namely 0—is called the limit value. In other words, when a sequence continues and its values get closer and closer to a specific reference point, that target value is called the limit value.
To summarize…
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Convergence → the phenomenon of getting closer to a certain value
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Limit → the concept used to describe that phenomenon
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Limit value → the reference value being approached
Why Do We Need the Concept of a Limit?
The reason we need limits is simple. They help us describe the behavior at the end. When a sequence or a changing value continues indefinitely, we want to know…
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Can its ending be described by a single number?
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Can we say that its behavior eventually becomes stable?
The concept of a limit allows us to answer these questions. In this sense, a limit is a tool used to determine whether the outcome of an ongoing process can be summarized by one number.
Expressing Limits with Mathematical Notation
So far, we have explained limits using words. Now let’s see how the same idea is expressed using mathematical notation. A limit is usually written in the following form
Although this expression may look complicated, it is simply a compact symbolic representation of what we have already discussed.
What This Expression Means
The equation above means
“As we continue the sequence , its values get closer and closer to a certain number .”
That is, the sequence converges to , and is the limit value.
Breaking Down the Symbols
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→ the value at the nth position in the sequence
(one of the numbers arranged in order)
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→ the sequence continues indefinitely
(∞ does not represent a number we reach, but the idea of continuing without end)
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→ a symbol that asks where the values approach as the sequence continues
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→ the reference value that the sequence approaches is $L$
Putting It Back into Words
This formula is simply a symbolic way of stating:
“As this sequence continues, its values get closer and closer to a certain number . Therefore, the limit value of this sequence is .”
To summarize everything discussed so far!!!
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Sequence → numbers arranged in a specific order
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Convergence → the phenomenon of getting closer to a certain value
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Divergence → the absence of a reference value that the sequence approaches
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Limit → the concept used to express the value being approached
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Limit value → the reference value itself
In short, a limit is a concept used to describe where a sequence is heading, using a single number.