answer:A local property in mathematics is a property that holds for a topological space if and only if it holds for every point in the space and every neighborhood around that point. It's not enough for the property to be true in some neighborhoods; it must also be the case that having the property in a neighborhood of each point implies that the entire space possesses the property. For instance, consider the property of continuity for a function f: X to Y between topological spaces X and Y. f is continuous if, for every x in X, there exists a neighborhood U such that the restriction f mid_U is continuous. This illustrates the local nature of continuity since we can check it pointwise in neighborhoods. Now, let's differentiate between a local property and a property being locally true. Being locally true means that a property holds in some neighborhoods, but it doesn't imply that the property holds globally for the entire space. An example of this is local connectivity: a topological space X is locally connected if every point has a neighborhood basis consisting of connected subspaces. However, X can be locally connected without being globally connected, as the property of being connected itself is not a local property. An example of a space that is locally characterized by a specific property without being globally characterized is a manifold. A manifold is a topological space that is locally homeomorphic to mathbb{R}^n, meaning that around every point, it can be mapped to an open subset of mathbb{R}^n in a continuous manner. However, not all manifolds are globally homeomorphic to mathbb{R}^n, emphasizing the distinction between local and global properties.

answer:To multiply a scalar (in this case, frac{1}{3}) with a matrix, you multiply each element of the matrix by the scalar: frac{1}{3} cdot left( begin{array}{cc} -3 & -8 1 & -6 end{array} right) = left( begin{array}{cc} frac{-3}{3} & frac{-8}{3} frac{1}{3} & frac{-6}{3} end{array} right) Simplify the fractions: left( begin{array}{cc} -1 & -frac{8}{3} frac{1}{3} & -2 end{array} right) Therefore, the product of the scalar frac{1}{3} and the matrix is: left( begin{array}{cc} -1 & -frac{8}{3} frac{1}{3} & -2 end{array} right)

answer:The author used a paradox, a statement that seems contradictory but contains a deeper truth.

answer:Let a_{n}= frac{(-3)^n}{(n+1) sqrt{n+1}} x^n. Then, left | frac{a_{n+1}}{a_{n}} right |=left | frac{(-3)^{n+1}}{(n+2) sqrt{n+2}} cdot x^{n+1}cdot frac{(n+1)sqrt{n+1}}{(-3)^{n}x^{n}} right | =left | left ( -3x right )frac{n+1}{n+2}sqrt{frac{n+1}{n+2}} right | =3left ( frac{n+1}{n+2} right )^{frac{3}{2}}left | x right |. Taking the limit as nrightarrow infty, we get: lim_{nrightarrow infty }left | frac{a_{n+1}}{a_{n}} right |=3left | x right |. The series converges absolutely for 3left | x right |< 1, or left | x right |< frac{1}{3}. When x=frac{1}{3}, the series becomes: sum_{n=1}^{infty }frac{(-1)^{n}}{(n+1)^{frac{3}{2}}}, which converges by the Alternating Series Test. When x=-frac{1}{3}, the series becomes: sum_{n=1}^{infty }frac{1}{(n+1)^{frac{3}{2}}}, which converges as it is a p-series with p=frac{3}{2}> 1. Therefore, the interval of convergence is left [ -frac{1}{3} ,frac{1}{3}right ].