Mathematics

p-series test criteria:

• According to the p-series convergence test, the infinite series $\sum_{n=1}^\infty \frac{1}{n^p}$ converges if and only if $p \gt 1$, and diverges if $p \le 1$. Among the given choices, only $p = 2$ satisfies $p \gt 1$.
• For $p = 1$, the series is the harmonic series $\sum \frac{1}{n}$, which diverges.
• For $p = 1/2$, $p \le 1$, so it diverges.
• For $p = 2/3$, $p \le 1$, so it diverges.

Analysis of p-series convergence:

• Both series are convergent, so neither is divergent.
• For $\sum \frac{1}{n^2}$, the exponent is $p = 2 \gt 1$, which converges by the p-series test. For $\sum \frac{1}{n^3}$, the exponent is $p = 3 \gt 1$, which also converges by the p-series test. Therefore, both series are convergent.
• Both are convergent.
• Both are convergent.

Fundamental theorem of skew-symmetric matrices:
By definition of skew-symmetric matrix: $A^T = -A$.
Taking the determinant of both sides:

Since $\det(A^T) = \det(A)$ and for an $n \times n$ matrix $\det(-A) = (-1)^n \det(A)$:

Here, the order $n = 5$ is an odd integer: