Understanding the Divergence of the Infinite Series -1/n^2

When analyzing the infinite series -1/n^2, one must carefully consider its behavior as n approaches infinity. The series in question is:

(sum_{n1}^{infty} -frac{1}{n^2})

It is important to note that this series is a negative variant of the well-known harmonic series. Most importantly, it diverges to negative infinity, meaning it does not converge to a finite sum. This implies that the sum of the series can become arbitrarily small (specifically, negative) as more terms are added.

Divergence and Partial Sums

The series diverges to negative infinity, and we can use partial sums to illustrate why. Consider the partial sum up to some large finite n:

(S_N -sum_{n1}^{N} frac{1}{n^2})

For sufficiently large N, this partial sum can be made smaller than any given real number. This is because the terms become increasingly small as n increases.

Asymptotic Convergence and Comparison

To understand the behavior of this series more rigorously, we can use asymptotic analysis. By comparing (-frac{1}{n^2}) to (-frac{1}{n}), we can apply asymptotic convergence:

(frac{-frac{1}{n^2}}{-frac{1}{n}} frac{1}{n})

As n approaches infinity, the limit of the ratio (lim_{n to infty} frac{1}{n} 0). This indicates that (-frac{1}{n^2}) behaves similarly to (-frac{1}{n}) for large n.

The harmonic series, which is (sum_{n1}^{infty} frac{1}{n}), is a well-known divergent series. Because (-frac{1}{n^2}) behaves similarly to (-frac{1}{n}) for large n, the series (sum_{n1}^{infty} -frac{1}{n^2}) also diverges to negative infinity.

Conclusion

Therefore, the given series diverges to negative infinity, which can be expressed as:

(sum_{n1}^{infty} -frac{1}{n^2} -infty)

Understanding this behavior is crucial in the study of infinite series, particularly when dealing with alternating series and their convergence properties.