Mathematics for Data Analysis and Modeling

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:

S(n) = (n/2) * (2a + (n-1)*d)

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:

Seq(i) = 1 / i, where i is the order of the element.

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.

But in fact, it's not so simple. Let's look at another sequence:

Seq(i) = 1 / i ** 2

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.

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 that a is greater than 1, the infinite sum of this sequence converges and is equal to some finite number.

Assume we have the following sequence S(i) = i^7 / i^8. Can we calculate the infinite sum of this sequence?

Select the correct answer

Yes, we can

No, because this sequence is equal to S(i) = 1 / i

Submit Answer