Bounding and Convergence in Series with Non-Negative Terms

When dealing with series with non-negative terms, a crucial and often used criterion for determining the series' convergence involves the properties of the sequence of its partial sums. In this article, we will delve into the intricacies of this criterion, understanding why and under what conditions the convergence of a series with non-negative terms is guaranteed by the boundedness of the sequence of its partial sums.

Introduction to Series Convergence and Partial Sums

A series is defined as the sum of the terms of an infinite sequence. It can be expressed as:

sum_{np}^{infty} a_n

where each term (a_n) is non-negative. The partial sums (s_N) are the sums of the first (N) terms of the series, defined as:

s_N  sum_{np}^{N} a_n

The series is said to converge if and only if the sequence of partial sums ( {s_N} ) converges to a finite limit.

Monotonic Increasing Sequence of Partial Sums

When the terms (a_n) are non-negative, an important property of the sequence of partial sums ( {s_N} ) emerges. Specifically, the sequence ( {s_N} ) is monotonically increasing. This means that for any two natural numbers (N_1

s_{N_1}  s_{N_2}

This is because adding more non-negative terms to the sum will only increase the sum itself, making the sequence of partial sums either constant or strictly increasing.

Boundedness and Convergence

A sequence is said to be bounded if there exists a real number (M) such that for all (N), the terms of the sequence satisfy:

|s_N|  M

In the case of the sequence of partial sums ( {s_N} ), being bounded implies that the sequence of partial sums never exceeds a certain value, no matter how large (N) gets. A fundamental theorem in analysis states that every monotonically increasing sequence of real numbers that is bounded above is convergent. Thus, if ( {s_N} ) is both monotonically increasing and bounded, then it must converge to a finite limit.

Implications for Convergence of Non-Negative Series

Combining the properties of monotonicity and boundedness, we can conclude that for a series with non-negative terms, the sequence of partial sums ( {s_N} ) is guaranteed to converge if and only if it is bounded. Mathematically, this can be stated as:

sum_{np}^{infty} a_n converges if and only if the sequence {s_N} is bounded

To illustrate this concept, consider the following example: let the series be (sum_{np}^{infty} a_n) where each (a_n geq 0). If we know that (s_N sum_{np}^{N} a_n) is bounded, then the series (sum_{np}^{infty} a_n) converges.

Counterexample: Alternating Series

While the above property is true for series with non-negative terms, it is important to note that it does not apply to series whose terms alternate in sign. For example, consider the series:

sum_{n0}^{infty} (-1)^n

This series has terms that alternate between 1 and -1. The sequence of partial sums for this series is:

s_N  1, 0, 1, 0, 1, 0, ...

Here, the partial sums oscillate between 1 and 0, thus forming a bounded sequence. However, the sequence of partial sums does not converge to a single finite value; instead, it oscillates indefinitely. This series is known as the Grandi's series, and it is a classic example of a conditionally convergent series where the sum depends on the order of summation.

Conclusion

In conclusion, for series with non-negative terms, the convergence is directly linked to the boundedness of the sequence of partial sums. This powerful property simplifies the analysis of many infinite series, especially those encountered in practical applications. However, caution is advised when dealing with series that have alternating terms, as their behavior can be much more complex and may not follow the same rules.

References

[1] Bartle, R. G., Sherbert, D. R. (2011). Introduction to Real Analysis (4th ed.). John Wiley Sons.

[2] Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill.