Chapter 30: Series
A Series is an expression consisting of a sequence of terms connected by the signs plus or minus. While we have studied finite series in previous chapters, we now turn our attention to Infinite Series and the conditions under which they represent a definite, finite value.
30.1 Convergent vs. Divergent Series
The most important characteristic of an infinite series is whether it "settles down" to a specific number or grows without bound.
- Convergent Series: A series where the sum of the first n terms approaches a fixed limit as n increases infinitely.
- Divergent Series: A series where the sum increases indefinitely, or does not approach a fixed limit, as n increases.
The Comparison Test: One of the simplest ways to tell if a series converges is to compare it term-by-term with a series we already know. If every term is smaller than the corresponding term of a known convergent series, the new series must also converge.
30.2 The Ratio Test
A more powerful tool is the Ratio Test. We examine the ratio of any term to the one preceding it as we go further out in the series.
If the limit of the ratio of the (n+1)th term to the nth term is less than 1, the series is convergent. If it is greater than 1, the series is divergent.
30.3 Power Series
A Power Series is a series of the form: c₀ + c₁x + c₂x² + c₃x³ + .... Whether such a series converges often depends on the value of x. The range of x-values for which the series converges is called the Interval of Convergence.
Example of Divergence: The "Harmonic Series" (1 + 1/2 + 1/3 + 1/4 + ...) is famously divergent. Even though the terms keep getting smaller, they don't get smaller fast enough to stop the total sum from eventually exceeding any number you can imagine!
Doctor's Pro-Tip: In practical engineering, we rarely need an "infinite" number of terms. We use the theory of series to determine how many terms we need to calculate to reach a desired level of precision—whether that's for a bridge truss or a radio frequency calculation.