Convergence of Series and Their Squares: A Comprehensive Guide

When dealing with infinite series, one often wonders about the behavior of the squares of the terms within these series. Specifically, if a series is convergent, is its square also convergent? This article will explore the conditions under which the square of a convergent series is also convergent and provide examples to illustrate these conditions.

Overview of Convergence of Series

A series is considered convergent if the sequence of its partial sums approaches a finite limit. In mathematical terms, a series (sum a_n) is convergent if and only if the sequence of partial sums (S_n sum_{i1}^{n} a_i) converges to a finite limit (L).

Convergence of the Square of a Series

Consider a convergent series (sum a_n). Will the series (sum a_n^2) also be convergent? The answer is not straightforward, as it depends on the specific nature of the series (sum a_n).

Absolutely Convergent Series

An important property to consider is absolute convergence. A series (sum a_n) is said to be absolutely convergent if the series of its absolute values (sum |a_n|) is convergent. If a series (sum a_n) is absolutely convergent, then it is indeed true that (sum a_n^2) is also convergent. This can be understood through the following reasoning:

The terms (a_n) must approach zero as (n) approaches infinity. (a_n^2) will approach zero faster, making (sum a_n^2) a more rapidly converging series.

To illustrate this, consider the series (sum frac{1}{n^2}). This series is known to be convergent, and its square series (sum left(frac{1}{n^2}right)^2 sum frac{1}{n^4}) is also convergent.

Conditionally Convergent Series

A series (sum a_n) is called conditionally convergent if it converges but the series of its absolute values (sum |a_n|) diverges. For such series, the convergence of (sum a_n^2) is not guaranteed.

Consider the alternating harmonic series (sum frac{(-1)^n}{n}). This series is well-known to be conditionally convergent. However, when squared, it results in the harmonic series (sum left(frac{1}{n}right)^2 sum frac{1}{n^2}), which is convergent. However, if we consider the series (sum frac{(-1)^n}{sqrt{n}}), this series is conditionally convergent, but the sum of the squares (sum left(frac{(-1)^n}{sqrt{n}}right)^2 sum frac{1}{n}) is the harmonic series, which is divergent.

Positive Terms Series

For a series with all positive terms, i.e., (a_n geq 0), the convergence of (sum a_n^2) is guaranteed if the original series (sum a_n) is convergent. This is because the terms (a_n^2) will also approach zero, and the convergence of a series of non-negative terms is guaranteed.

However, if the series contains negative terms, the situation can be more complex. For example, the series (sum frac{(-1)^n}{sqrt{n}}) converges, but the series (sum left(frac{(-1)^n}{sqrt{n}}right)^2 sum frac{1}{n}) diverges.

Conclusion

In summary, the convergence of a series (sum a_n) implies the convergence of (sum a_n^2) in the case of absolute convergence. However, for conditionally convergent series, the convergence of (sum a_n^2) is not guaranteed. These properties provide a deeper understanding of the behavior of series and their squares.