Learn Introduction to Series

Definition

A series is a mathematical expression formed by adding the terms of a sequence. The most common types are the arithmetic series and the geometric series, which differ in how their terms progress.

Arithmetic Series

An arithmetic series is formed when the difference between consecutive terms in a sequence is constant.

2, 5, 8, 11, 14, ...; (\text{common difference}, d = 3)

The sum of the first $n$ terms of an arithmetic series is given by:

S_n = \frac{n}{2} \cdot (a + l)

Where:

$n$ - number of terms;
$a$ - first term;
$l$ - last term.

Alternatively, if the last term $l$ is not known:

S_n = \frac{n}{2} \cdot 2a + (n - 1) \cdot d

Example

Find the sum of the first 10 terms of the series $2,5,8,...$

S_{10} = \frac{10}{2} \cdot (2 + (10 - 1) \cdot 3) = 5 \cdot (2 + 27) = 145

Geometric Series

A geometric series is formed when each term in the sequence is multiplied by a fixed ratio to get the next term.

3,6,12,24,48,...;(\text{common ratio}, r=2)

The sum of the first $n$ terms of a geometric series is given by:

S_n = a \cdot \frac{1 - r^n}{1 - r},\ r \neq 1

Where:

$a$ - first term;
$r$ - common ratio;
$n$ - number of terms.

If the series is infinite and $|r|<1$ :

S = \frac{a}{1 - r}

Example:

Find the sum of the first 4 terms of the series $3,6,12,24,...$

S_4 = 3 \cdot \frac{1-2^4}{1-2} = 3 \cdot \frac{1-16}{-1}=3 \cdot 15 = 45

Real-World Applications

Arithmetic and geometric series appear in many data science contexts:

Population growth and resource modeling through geometric progressions;
Financial analysis using compound interest calculations;
Revenue forecasting across time periods;
Machine learning, where summations occur in algorithms like gradient descent.

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Section 2. Chapter 4

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Definition

Arithmetic Series

An arithmetic series is formed when the difference between consecutive terms in a sequence is constant.

2, 5, 8, 11, 14, ...; (\text{common difference}, d = 3)

The sum of the first $n$ terms of an arithmetic series is given by:

S_n = \frac{n}{2} \cdot (a + l)

Where:

$n$ - number of terms;
$a$ - first term;
$l$ - last term.

Alternatively, if the last term $l$ is not known:

S_n = \frac{n}{2} \cdot 2a + (n - 1) \cdot d

Example

Find the sum of the first 10 terms of the series $2,5,8,...$

S_{10} = \frac{10}{2} \cdot (2 + (10 - 1) \cdot 3) = 5 \cdot (2 + 27) = 145

Geometric Series

A geometric series is formed when each term in the sequence is multiplied by a fixed ratio to get the next term.

3,6,12,24,48,...;(\text{common ratio}, r=2)

The sum of the first $n$ terms of a geometric series is given by:

S_n = a \cdot \frac{1 - r^n}{1 - r},\ r \neq 1

Where:

$a$ - first term;
$r$ - common ratio;
$n$ - number of terms.

If the series is infinite and $|r|<1$ :

S = \frac{a}{1 - r}

Example:

Find the sum of the first 4 terms of the series $3,6,12,24,...$

S_4 = 3 \cdot \frac{1-2^4}{1-2} = 3 \cdot \frac{1-16}{-1}=3 \cdot 15 = 45

Real-World Applications

Arithmetic and geometric series appear in many data science contexts:

Population growth and resource modeling through geometric progressions;
Financial analysis using compound interest calculations;
Revenue forecasting across time periods;
Machine learning, where summations occur in algorithms like gradient descent.

Everything was clear?

Thanks for your feedback!

Section 2. Chapter 4