Convergence and Uniform Convergence of the Series (sum_{n1}^{infty} nx^{-2})

The analysis of the convergence and uniform convergence of the series (sum_{n1}^{infty} nx^{-2}) is a fundamental topic in mathematical analysis. In this article, we will break down the process step by step, examining both pointwise and uniform convergence of this series.

Step 1: Convergence of the Series

Convergence of the Inner Series

The inner series (sum_{n1}^{infty} frac{1}{n^2}) is a well-known convergent series, known as the Basel problem. This series converges to (frac{pi^2}{6}).

Convergence of the Original Series

Therefore, for (x eq 0), the series (sum_{n1}^{infty} nx^{-2} sum_{n1}^{infty} frac{1}{n^2 x^2} frac{1}{x^2} sum_{n1}^{infty} frac{1}{n^2}) converges for any (x eq 0).

Step 2: Uniform Convergence

To analyze uniform convergence, we consider the series as a function of (x): (f(x) sum_{n1}^{infty} frac{1}{n^2 x^2}). For an interval (I) where (x) is bounded away from zero, say (I [a, b]) with (0 Pointwise Convergence: For each fixed (x in I), the series converges to (f(x) frac{1}{x^2} cdot frac{pi^2}{6}). Uniform Convergence: To show uniform convergence, we use the Weierstrass M-test. We need to find a constant (M_n) such that: (left|frac{1}{n^2 x^2}right| leq M_n) Since (x) is bounded away from zero, there exists a constant (m min_{x in I} x > 0). Thus,(left|frac{1}{n^2 x^2}right| leq frac{1}{n^2 m^2}). The series (sum_{n1}^{infty} frac{1}{n^2 m^2} frac{1}{m^2} sum_{n1}^{infty} frac{1}{n^2}) converges since (sum_{n1}^{infty} frac{1}{n^2}) converges. By the Weierstrass M-test, we conclude that the series converges uniformly on any interval (I) where (x) is bounded away from zero.

Conclusion

The series (sum_{n1}^{infty} nx^{-2}) converges for all (x eq 0).

The series converges uniformly on any interval (I) where (x) is bounded away from zero, i.e., (x in [a, b]) with (0

Additional Considerations

First, it is evident that the terms in the series are undefined at (x 0). Therefore, we must examine the convergence over one of the open half-lines or half-closed half-lines for some positive (a). Since all terms are even functions, it suffices to consider the positive half-lines.

Given that the numerical series (sum_{n1}^{infty} frac{1}{n^2}) is absolutely convergent to a finite limit, it follows that the series converges pointwise at every non-zero point (x). Moreover, convergence is uniform over half-closed half-lines ([a, b]) for any choice of (a > 0) because the tails of the series tend to zero as (N) tends to infinity.

However, when considering open half-lines, the convergence is not uniform. The term (frac{1}{x^2}) tends to infinity as (x) approaches zero, and for any natural number (N), we can find (x) non-zero but close enough to (0) such that the terms are still significant, indicating that the convergence of the series is not uniform over open half-lines.