Series

Mathematics > Real Analysis > Series

A series in the context of real analysis is a concept that encompasses the summation of elements of a sequence. Specifically, a series is defined as the sum of a sequence of real numbers. Given a sequence \( \{a_n\} \), the corresponding series is commonly expressed as:

\[ S = \sum_{n=1}^{\infty} a_n. \]

In simple terms, a series is the cumulative addition of the terms of a sequence, and it provides a tool for investigating the properties of infinities and limits in a rigorous way.

Convergence and Divergence

One of the central themes in the study of series is determining whether the series converges or diverges. A series \(\sum_{n=1}^{\infty} a_n\) is said to converge if the sequence of partial sums \( S_N \) defined by

\[ S_N = \sum_{n=1}^{N} a_n \]

has a finite limit as \( N \) approaches infinity. That is,

\[ \lim_{N \to \infty} S_N = S, \]

where \( S \) is a real number. If such a limit does not exist or is infinite, the series is said to diverge.

Tests for Convergence

There are several methods to test the convergence of a series. Some of the most notable include:

1. The Comparison Test:
   If \( 0 \leq a_n \leq b_n \) for all \( n \) and \( \sum_{n=1}^{\infty} b_n \) is known to converge, then \( \sum_{n=1}^{\infty} a_n \) also converges.

2. The Ratio Test:
   Consider a series \(\sum_{n=1}^{\infty} a_n \) and assume that \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \). Then:

   • If \( L < 1 \), the series converges absolutely.
   • If \( L > 1 \), the series diverges.
   • If \( L = 1 \), the test is inconclusive.

3. The Root Test:
   For a series \(\sum_{n=1}^{\infty} a_n \), define \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \). Then:

   • If \( L < 1 \), the series converges absolutely.
   • If \( L > 1 \), the series diverges.
   • If \( L = 1 \), the test is inconclusive.

4. The Integral Test:
   If \( f(n) = a_n \) is a positive, decreasing, and continuous function, and \( \int_{1}^{\infty} f(x) \, dx \) converges, then so does \(\sum_{n=1}^{\infty} a_n \).

5. Alternating Series Test (Leibniz’s Test):
   For an alternating series of the form \( \sum_{n=1}^{\infty} (-1)^{n-1} b_n \), if the sequence \( \{b_n\} \) is monotonically decreasing and \( \lim_{n \to \infty} b_n = 0 \), then the series converges.

Types of Convergence

• Absolute Convergence:
  A series \(\sum_{n=1}^{\infty} a_n \) converges absolutely if \(\sum_{n=1}^{\infty} |a_n| \) converges.

• Conditional Convergence:
  A series \(\sum_{n=1}^{\infty} a_n \) converges conditionally if it converges, but does not converge absolutely.

Important Series

Several well-known series serve as foundational examples and are frequently encountered in real analysis:

• Geometric Series:
  \[ \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \quad \text{for} \quad |r| < 1. \]

• Harmonic Series:
  \[ \sum_{n=1}^{\infty} \frac{1}{n}, \]
  which diverges.

• p-Series:
  \[ \sum_{n=1}^{\infty} \frac{1}{n^p}. \]
  The series converges if \( p > 1 \) and diverges if \( p \leq 1 \).

In summary, the study of series in real analysis provides a robust framework for understanding infinite summations and their convergence properties. It is a critical tool in mathematical analysis with applications extending to various areas of mathematics, physics, and engineering.