An infinite series is the sum of infinitely many terms of a sequence. It is written in the form:

\[
S = a_1 + a_2 + a_3 + \cdots = \sum_{n=1}^{\infty} a_n,
\]

where \( a_n \) represents the terms of the sequence. The formula for an infinite series can vary depending on the type of series. Common types of infinite series include geometric series, arithmetic series, harmonic series, and more.

### 1. Geometric Series
A geometric series is one where each term is a constant multiple (called the common ratio, \( r \)) of the previous term. The infinite geometric series with the first term \( a \) and common ratio \( r \) (where \( |r| < 1 \)) can be expressed as:

\[
S = a + ar + ar^2 + ar^3 + \cdots = \sum_{n=0}^{\infty} ar^n.
\]

**Formula for the sum of an infinite geometric series:**

\[
S = \frac{a}{1 - r}, \quad \text{for } |r| < 1.
\]

#### Example of an Infinite Geometric Series:
Consider the series: 

\[
S = 3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \cdots
\]

Here, the first term \( a = 3 \), and the common ratio \( r = \frac{1}{2} \).

Since \( |r| = \frac{1}{2} < 1 \), the series converges, and we can find its sum:

\[
S = \frac{a}{1 - r} = \frac{3}{1 - \frac{1}{2}} = \frac{3}{\frac{1}{2}} = 6.
\]

So, the sum of this infinite geometric series is 6.

### 2. Harmonic Series
The harmonic series is an example of a divergent infinite series. It is defined as:

\[
H = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots = \sum_{n=1}^{\infty} \frac{1}{n}.
\]

The harmonic series does not converge; its sum grows without bound as more terms are added.

#### Example of a Harmonic Series:
The infinite series:

\[
1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots
\]

does not have a finite sum, i.e., it diverges.

### 3. Power Series
A power series is an infinite series in which each term is a power of a variable \( x \). It is of the form:

\[
\sum_{n=0}^{\infty} c_n x^n,
\]

where \( c_n \) are coefficients and \( x \) is a variable.

#### Example of a Power Series:
The Maclaurin series for the exponential function \( e^x \) is:

\[
e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}.
\]

### Conclusion
Infinite series have a broad range of applications in mathematics, particularly in calculus, analysis, and applied mathematics. Some series converge to a finite value, while others do not.

Would you like further details or examples on a specific type of infinite series?

### Relative Questions:
1. What is the difference between a convergent and divergent series?
2. How can we determine if a series converges or diverges?
3. What are some real-life applications of infinite series?
4. Can you explain the concept of a Taylor series?
5. How are infinite series used in calculus?

**Tip:** For infinite geometric series, make sure that the common ratio \( r \) is between -1 and 1 for the series to converge to a finite sum.

Explore infinite series, including geometric, harmonic, and power series. Learn how to calculate sums of infinite geometric series with examples and formulas. Suitable for advanced mathematics students.

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