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:


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def sum_arithmetic_progression_formula(a, d, n):
    # Calculate the sum of an arithmetic progression using the formula
    sum_formula = (n / 2) * (2 * a + (n - 1) * d)  # Sum formula for arithmetic progression
    return sum_formula

def sum_arithmetic_progression_loop(a, d, n):
    # Calculate the sum of an arithmetic progression using a loop
    progression_sum = 0  # Initialize the sum of the arithmetic progression
    current_term = a  # Set the current term to the first term
    for _ in range(n):  # Loop through each term in the arithmetic progression
        progression_sum += current_term  # Add the current term to the sum
        current_term += d  # Move to the next term by adding the common difference
    return progression_sum

# Example usage
a = 2  # First term
d = 3  # Common difference
n = 5  # Number of terms in the arithmetic progression

# Calculate the sum using the formula
sum_formula = sum_arithmetic_progression_formula(a, d, n)
print(f'Sum using formula is {sum_formula}')

# Calculate the sum using a loop
sum_loop = sum_arithmetic_progression_loop(a, d, n)
print(f'Sum using loop is {sum_loop}')

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.


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def calculate_sequence_sum():
    # Initialize variables
    sequence_sum = 0  # Initialize the sum of the sequence
    partial_sums = []  # Initialize a list to store partial sums

    # Loop through elements in the sequence
    for n in range(1, 10001):
        term = 1 / n  # Calculate the term of the sequence
        sequence_sum += term  # Add the term to the sequence sum

        # Check if current index is a multiple of 1000
        if n % 1000 == 0:
            partial_sums.append(sequence_sum)  # Store the partial sum
            # Print partial sum up to the current index
            print(f'Partial sum of elements 1 to {n}: {sequence_sum:.5f}')

# Call the function to calculate the sequence sum
calculate_sequence_sum()

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:


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def calculate_sequence_sum():

    sequence_sum = 0
    partial_sums = []

    for n in range(1, 10001):
        term = 1 / (n ** 2)
        sequence_sum += term

        if n % 1000 == 0:
            partial_sums.append(sequence_sum)
            print(f'Partial sum of elements 1 to {n}: {sequence_sum:.5f}')


calculate_sequence_sum()

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.

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