Geometric Series

• Formula for the Sum of a Geometric Series
  • Formula for the Sum of the First $n$ Terms:
  • Example:
• Convergence of Geometric Series
  • Infinite Geometric Series:
  • Example:
  • Divergence of Geometric Series:
  • Example of Divergence:
• Applications of Geometric Series
  • 1. Finance: Calculating Compound Interest
    • Example:
  • 2. Physics: Motion and Damping
    • Example:
  • 3. Computer Science: Algorithm Analysis
    • Example:
  • 4. Signal Processing: Fourier Series

Formula for the Sum of a Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio $r$. The general form of a geometric series is:

Where:

• $a$ is the first term.
• $r$ is the common ratio (the factor by which each term is multiplied to get the next term).

Formula for the Sum of the First $n$ Terms:

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

Where:

• $S_n$ is the sum of the first $n$ terms.
• $a$ is the first term.
• $r$ is the common ratio.
• $n$ is the number of terms.

Example:

Find the sum of the first 5 terms of the geometric series where $a=2$ and $r=3$:

Convergence of Geometric Series

Infinite Geometric Series:

An infinite geometric series is one where the number of terms approaches infinity. The sum of an infinite geometric series is given by the limit of the partial sums as $n\to \mathrm{\infty }$.

If $|r|<1$, the infinite geometric series converges, and the sum is:

If $|r|\ge 1$, the series diverges.

Example:

Find the sum of the infinite geometric series $S=3+3\cdot \frac{1}{2}+3\cdot {\frac{1}{2}}^2+3\cdot {\frac{1}{2}}^3+\dots$

• Here, $a=3$ and $r=\frac{1}{2}$.
• Since $|r|=\frac{1}{2}<1$, the series converges.
• The sum is:

Divergence of Geometric Series:

If $|r|\ge 1$, the geometric series diverges. This means that the sum grows without bound as the number of terms increases, or the series does not approach a finite value.

Example of Divergence:

The geometric series $S=1+2+4+8+\dots$ with $a=1$ and $r=2$ diverges because $|r|=2\ge 1$.

Applications of Geometric Series

1. Finance: Calculating Compound Interest

Geometric series are used in finance to calculate the future value of investments with compound interest.

Example:

Consider an investment with an initial amount $P$, annual interest rate $r$, compounded annually for $n$ years. The total value $A$ of the investment after $n$ years is:

This formula comes from summing the powers of $(1+r)$ over time, which is a geometric series.

2. Physics: Motion and Damping

Geometric series arise in physics when studying systems with exponential decay or damping, where successive displacements or amplitudes form a geometric sequence.

Example:

A ball dropped from a height that bounces back to half its height each time forms a geometric series. The total distance the ball travels before coming to rest is:

Where $h$ is the initial height.

3. Computer Science: Algorithm Analysis

Geometric series are used in computer science to analyze the time complexity of recursive algorithms, particularly those that involve dividing a problem into smaller subproblems.

Example:

In the merge sort algorithm, each recursive division step splits the input array in half, forming a geometric series in terms of time complexity.

4. Signal Processing: Fourier Series

Geometric series are applied in signal processing to represent periodic functions as sums of sines and cosines, enabling frequency analysis in signals.