In calculus, we study sequences to help us understand series.
Think of a sequence as an ordered list of numbers.
A sequence can be written as a function that uses natural numbers (like 1, 2, 3, and so on).
We often write a sequence like this: ( {a_n} ). Here, ( n ) is the order in the sequence, and ( a_n ) is the number in that place.
We can express a sequence using a formula. For example, the formula ( a_n = \frac{1}{n} ) describes the harmonic series.
Another example is the Fibonacci sequence, which we can write as ( F_n = F_{n-1} + F_{n-2} ). The first two numbers in this sequence are ( F_0 = 0 ) and ( F_1 = 1 ).
One important idea when studying sequences is convergence.
A sequence ( {a_n} ) converges to a limit ( L ) if, for every tiny number ( \epsilon > 0 ), there is a positive integer ( N ) such that for all ( n ) larger than ( N ), the difference between ( a_n ) and ( L ) is less than ( \epsilon ).
This means that as ( n ) gets bigger, the sequence gets closer and closer to a specific value.
For instance, the sequence ( \left{ \frac{1}{n} \right} ) converges to 0 because, as ( n ) grows larger, the numbers get very close to 0.
On the other hand, a sequence diverges if it doesn’t settle down to any one number.
Divergence can happen in different ways: the numbers might keep getting bigger and bigger, switch between values, or approach different numbers.
A common example of a divergent sequence is ( {(-1)^n} ), which keeps switching between -1 and 1.
We can also sort sequences by how they behave.
A sequence is called monotonic if it always goes up or always goes down.
Specifically, a sequence ( {a_n} ) is increasing if ( a_n \leq a_{n+1} ) for all ( n ).
It's important to know that every monotonic sequence that is bounded (meaning it doesn’t go beyond certain limits) will converge.
In math writing and graphs, we often show the terms of a sequence like this: ( S = {a_1, a_2, a_3, \ldots} ).
We sometimes express the limit of a sequence as ( \lim_{n \to \infty} a_n = L ).
Finally, when we look at series, we use something called the n-th term test to decide if they diverge.
If the limit of the sequence ( {a_n} ) is not 0 (meaning ( \lim_{n \to \infty} a_n \neq 0 )), then the series ( \sum_{n=1}^{\infty} a_n ) diverges.
By understanding sequences—their behavior, if they converge, and how we write them—we set the groundwork to learn more about series and their applications in calculus.
In calculus, we study sequences to help us understand series.
Think of a sequence as an ordered list of numbers.
A sequence can be written as a function that uses natural numbers (like 1, 2, 3, and so on).
We often write a sequence like this: ( {a_n} ). Here, ( n ) is the order in the sequence, and ( a_n ) is the number in that place.
We can express a sequence using a formula. For example, the formula ( a_n = \frac{1}{n} ) describes the harmonic series.
Another example is the Fibonacci sequence, which we can write as ( F_n = F_{n-1} + F_{n-2} ). The first two numbers in this sequence are ( F_0 = 0 ) and ( F_1 = 1 ).
One important idea when studying sequences is convergence.
A sequence ( {a_n} ) converges to a limit ( L ) if, for every tiny number ( \epsilon > 0 ), there is a positive integer ( N ) such that for all ( n ) larger than ( N ), the difference between ( a_n ) and ( L ) is less than ( \epsilon ).
This means that as ( n ) gets bigger, the sequence gets closer and closer to a specific value.
For instance, the sequence ( \left{ \frac{1}{n} \right} ) converges to 0 because, as ( n ) grows larger, the numbers get very close to 0.
On the other hand, a sequence diverges if it doesn’t settle down to any one number.
Divergence can happen in different ways: the numbers might keep getting bigger and bigger, switch between values, or approach different numbers.
A common example of a divergent sequence is ( {(-1)^n} ), which keeps switching between -1 and 1.
We can also sort sequences by how they behave.
A sequence is called monotonic if it always goes up or always goes down.
Specifically, a sequence ( {a_n} ) is increasing if ( a_n \leq a_{n+1} ) for all ( n ).
It's important to know that every monotonic sequence that is bounded (meaning it doesn’t go beyond certain limits) will converge.
In math writing and graphs, we often show the terms of a sequence like this: ( S = {a_1, a_2, a_3, \ldots} ).
We sometimes express the limit of a sequence as ( \lim_{n \to \infty} a_n = L ).
Finally, when we look at series, we use something called the n-th term test to decide if they diverge.
If the limit of the sequence ( {a_n} ) is not 0 (meaning ( \lim_{n \to \infty} a_n \neq 0 )), then the series ( \sum_{n=1}^{\infty} a_n ) diverges.
By understanding sequences—their behavior, if they converge, and how we write them—we set the groundwork to learn more about series and their applications in calculus.