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Decimal expansion refers to representing numbers as a sum of fractions. These fractions have powers of ten in the denominator. For example, the decimal number 0.abc... means each digit holds a value based on its position. Hence, it becomes:

• \( a/10 \) for the first digit after the decimal point,
• \( b/100 \) for the second digit,
• \( c/1000 \) for the third digit, and so forth.

In essence, a decimal expansion is just another way of writing an infinite series, where every digit contributes in descending order. The beauty of decimal expansions lies in their ability to represent any real number, whether rational or irrational, through an infinite sum of powers of 10.

The comparison test is a technique to determine if a series is convergent. It relies on comparing a series you are investigating with a known one. If the series in question has terms that are always smaller than those of a known convergent series, it too converges. Here's how it works:

• Identify the series you wish to analyze. For decimal expansions, it's \( \sum_{n=1}^{\infty} \frac{d_n}{10^n} \).
• Select a known convergent series, like the simple geometric series \( \sum_{n=1}^{\infty} \frac{1}{10^n} \).
• Show that each term in your series is less than or equal to the corresponding term in the known series.
• If the series of larger terms converges, then by the Comparison Test, your original series also converges.

This approach is part of a suite of tools for working with infinite series. It makes working with complicated series manageable by referring to simpler, well-studied ones.

A geometric series is a series of numbers where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The classic form of a geometric series is:

• \( a, ar, ar^2, ar^3, \ldots \)

The sum of an infinite geometric series forms when the absolute value of the ratio is less than one. The formula to find the sum is: \[S = \frac{a}{1 - r}\] The series \( \sum_{n=1}^{\infty} \frac{1}{10^n} \) serves as an example where a convergent geometric series with ratio \( r = 1/10 \). Here, each term gets progressively smaller, making the series sum finite. Recognizing such series is crucial when comparing them to others, especially in proofs of convergence using methods like the Comparison Test.