Series Convergence : Prove that if an ≥ 0 and Σ an converges, then Σ(√an / n) converges.
Prove that if an ≥ 0 and Σ an converges, then Σ(√an / n) converges. Please see the attached file for the fully formatted problem.
Integral (or Block) Test for Convergence
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Characteristic Function of Metric Space
Let S ⊂ M. (a) Define the characteristic function Xs : M --> R. (b) If M is a metric space, show that Xs(x) is discontinuous at x if and only if x is a boundary point of S. [Please see attached PDF file for full problem]. for part (a), I think something similar to http://planetmath.org/encyclopedia/Characteristi ...continues
Banach Space : Norm in Dual Space
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Real Analysis : Riemann Stieltjes Integral
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Fourier Analysis and Series : Periodic Functions and Convolution
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Suppose that f_k -> f uniformly on (0,1). Let 0 < x < 1. Suppose that lim f_k(t) = A_k for k=1,2,... Show that {A_k} converges and lim f(t) = LIM A_k. That is show lim LIM f_k(t) = LIM lim f_k(t). Where lim represents the limit as t approaches x and LIM represents the limit as k approaches infinity.
Topological Spaces and Continuity
Please see the attached file for the fully formatted problems. Let be a real valued function on a topological space . Show that is continuous if and only if for each real number the set and are open. Show that is continuous if and only if for each real number the set is open and is closed ...continues
Scatterplots, Least Squares Regression, Coefficient of Determination and Forecasting
a) Construct a scatter diagram of the number of crimes and police expenditures per capita, with number of crimes as the predictor variable. What can you say about the relationship between these two variables based on the scatter plot? b) Find the least-squares regression equation that predicts police expenditures per capita fro ...continues
Show that f is proper if and only if f* is continuous
Let X and Y be locally compact Hausdorff spaces. Let X* and Y* be their one point compactifications. Let f be a continuous map from X to Y. Let f* be the map from X* to Y* whose restriction to X is f, and which takes the point at infinity in X* to the point at infinity in Y*. Show that f is proper if and only if f* is contin ...continues
