Contenido del Curso
Mathematics for Data Analysis and Modeling
1. Basic Mathematical Concepts and Definitions
2. Linear Algebra
Mathematics for Data Analysis and Modeling
Sum of Elements of The Series
What is the sum of the series?
The sum of a series refers to adding up the values in the sequence. There are two situations to consider: finding the sum of a finite number of elements or the sum of all elements in the sequence. Depending on the situation, different approaches are used to calculate the sum.
Finite sums
If we work with finite sums, we can loop through all the elements to calculate the sum. Also, some types of sequences have ready-made formulas for calculating the sum.
For example, the sum of n elements of arithmetic progression can be calculated as follows:
Infinite sums
In the case of summing absolutely all elements of the sequence, we will encounter certain problems. For example, consider a sequence like this:
Let's look at the sum of 10000 first elements of the sequence. We will also print partial sums of 1000, 2000, 3000, etc. elements.
We can see that partial sums are constantly increasing, and thus we cannot calculate the sum of all elements - it will be infinite. This conclusion seems quite logical - the more elements we summarize, the larger the sum. But since, formally, this sequence can be continued for an arbitrarily long time, the sum will be arbitrarily large.
In finance, this sequence is used to understand how interest affects savings over time. It helps in calculating how quickly your money grows when interest is added more frequently, like monthly or daily.
But in fact, it's not so simple. Let's look at another sequence:
Here we see that the partial sums are identical up to 3 digits, and the sum does not grow so rapidly with increased elements.
This is explained by the fact that elements with a large order are extremely small and eventually cease to affect the final sum significantly.
In physics, this sequence is used to describe the intensity of light or sound waves at different distances from a source. This is known as the inverse square law. For example, it explains why the brightness of a light or the loudness of a sound decreases rapidly as you move away from the source.
Convergent sums of the series
Thus, there is a certain class of sequences for which we can calculate the sum of absolutely all elements, which will be finite. This property is also called the convergence of the sum of a number series.
Note
In practice, the sum of the following sequence is often used:
S(i) = 1 / i^a. Provided thatais greater than1, the infinite sum of this sequence converges and is equal to some finite number.
¿Todo estuvo claro?
Contenido del Curso
Mathematics for Data Analysis and Modeling
1. Basic Mathematical Concepts and Definitions
2. Linear Algebra
Mathematics for Data Analysis and Modeling
Sum of Elements of The Series
What is the sum of the series?
The sum of a series refers to adding up the values in the sequence. There are two situations to consider: finding the sum of a finite number of elements or the sum of all elements in the sequence. Depending on the situation, different approaches are used to calculate the sum.
Finite sums
If we work with finite sums, we can loop through all the elements to calculate the sum. Also, some types of sequences have ready-made formulas for calculating the sum.
For example, the sum of n elements of arithmetic progression can be calculated as follows:
Infinite sums
In the case of summing absolutely all elements of the sequence, we will encounter certain problems. For example, consider a sequence like this:
Let's look at the sum of 10000 first elements of the sequence. We will also print partial sums of 1000, 2000, 3000, etc. elements.
We can see that partial sums are constantly increasing, and thus we cannot calculate the sum of all elements - it will be infinite. This conclusion seems quite logical - the more elements we summarize, the larger the sum. But since, formally, this sequence can be continued for an arbitrarily long time, the sum will be arbitrarily large.
In finance, this sequence is used to understand how interest affects savings over time. It helps in calculating how quickly your money grows when interest is added more frequently, like monthly or daily.
But in fact, it's not so simple. Let's look at another sequence:
Here we see that the partial sums are identical up to 3 digits, and the sum does not grow so rapidly with increased elements.
This is explained by the fact that elements with a large order are extremely small and eventually cease to affect the final sum significantly.
In physics, this sequence is used to describe the intensity of light or sound waves at different distances from a source. This is known as the inverse square law. For example, it explains why the brightness of a light or the loudness of a sound decreases rapidly as you move away from the source.
Convergent sums of the series
Thus, there is a certain class of sequences for which we can calculate the sum of absolutely all elements, which will be finite. This property is also called the convergence of the sum of a number series.
Note
In practice, the sum of the following sequence is often used:
S(i) = 1 / i^a. Provided thatais greater than1, the infinite sum of this sequence converges and is equal to some finite number.
¿Todo estuvo claro?
