Difference between revisions of "Talk:Aufgaben:Problem 15"
From Ferienserie MMP2
| Line 1: | Line 1: | ||
| + | Let \(t>0\) fixed. Define \( f(x) := x^{t-1}e^{-x} \), non-negative function on (0,\(\infty\)). | ||
==Part a)== | ==Part a)== | ||
| + | ===Problem=== | ||
| + | For \(k = 1,2,... \) let $$ f_k(x) = \left\{ \begin{array}{l l} x^{t-1}(1-\frac{x}{k})^{k} & \quad 0<x<k\\ 0 & \quad k\leq x<\infty\end{array} \right. $$ | ||
| + | Show that \(f_k(x) \rightarrow f(x) \) and \(f_k(x) \leq f_{k+1}(x), \forall x>0.\) | ||
| + | Hint: cjivo | ||
===Solution=== | ===Solution=== | ||
==Part b)== | ==Part b)== | ||
| + | ===Problem=== | ||
| + | |||
===Solution=== | ===Solution=== | ||
Revision as of 15:19, 28 December 2014
Let \(t>0\) fixed. Define \( f(x) := x^{t-1}e^{-x} \), non-negative function on (0,\(\infty\)).
Part a)
Problem
For \(k = 1,2,... \) let $$ f_k(x) = \left\{ \begin{array}{l l} x^{t-1}(1-\frac{x}{k})^{k} & \quad 0<x<k\\ 0 & \quad k\leq x<\infty\end{array} \right. $$ Show that \(f_k(x) \rightarrow f(x) \) and \(f_k(x) \leq f_{k+1}(x), \forall x>0.\) Hint: cjivo
