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Kenneth Rosen Edition 7 Exercise 2.5 Question 8 (Page No. 176)
Show that a countably infinite number of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
Show that a countably infinite number of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
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486
Kenneth Rosen Edition 7 Exercise 2.5 Question 6 (Page No. 176)
Suppose that Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel.
Suppose that Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel.
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Kenneth Rosen Edition 7 Exercise 2.5 Question 5 (Page No. 176)
Show that a finite group of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
Show that a finite group of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
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495
Kenneth Rosen Edition 7 Exercise 2.4 Question 43 (Page No. 170)
What are the values of the following products? $\displaystyle{}\prod_{i=0}^{10} i$ $\displaystyle{}\prod_{i=5}^{8} i$ $\displaystyle{}\prod_{i=1}^{100} (-1)^{i}$ $\displaystyle{}\prod_{i=1}^{10} 2$
What are the values of the following products?$\displaystyle{}\prod_{i=0}^{10} i$$\displaystyle{}\prod_{i=5}^{8} i$$\displaystyle{}\prod_{i=1}^{100} (-1)^{i}$$\displaysty...
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497
Kenneth Rosen Edition 7 Exercise 2.4 Question 41 (Page No. 169)
Find a formula for $\displaystyle{}\sum_{k=0}^{m}\left \lfloor \sqrt{k} \right \rfloor ,$ when $m$ is a positive integer.
Find a formula for $\displaystyle{}\sum_{k=0}^{m}\left \lfloor \sqrt{k} \right \rfloor ,$ when $m$ is a positive integer.
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502
Kenneth Rosen Edition 7 Exercise 2.4 Question 35 (Page No. 169)
Show that $\displaystyle{}\sum_{j=1}^{n}(a_{j} - a_{j-1}) = a_{n} -a_{0,}$ where $a_{0}, a_{1},\dots,a_{n}$ is a sequence of real numbers. This type of sum is called telescoping.
Show that $\displaystyle{}\sum_{j=1}^{n}(a_{j} - a_{j-1}) = a_{n} -a_{0,}$ where $a_{0}, a_{1},\dots,a_{n}$ is a sequence of real numbers. This type of sum is called tele...
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507
Kenneth Rosen Edition 7 Exercise 2.4 Question 30 (Page No. 169)
What are the values of these sums, where $S = \{1, 3, 5, 7\}?$ $\displaystyle{}\sum_{j\in S}j$ $\displaystyle{}\sum_{j\in S}j^{2}$ $\displaystyle{}\sum_{j\in S}(1/j)$ $\displaystyle{}\sum_{j\in S}1$
What are the values of these sums, where $S = \{1, 3, 5, 7\}?$$\displaystyle{}\sum_{j\in S}j$$\displaystyle{}\sum_{j\in S}j^{2}$$\displaystyle{}\sum_{j\in S}(1/j)$$\displ...
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508
Kenneth Rosen Edition 7 Exercise 2.4 Question 29 (Page No. 169)
What are the values of these sums? $\displaystyle{}\sum_{k=1}^{5}(k+1)$ $\displaystyle{}\sum_{j=0}^{4}(-2)^{j}$ $\displaystyle{}\sum_{i=1}^{10}3$ $\displaystyle{}\sum_{j=0}^{8}(2^{j+1}-2^{j})$
What are the values of these sums?$\displaystyle{}\sum_{k=1}^{5}(k+1)$$\displaystyle{}\sum_{j=0}^{4}(-2)^{j}$$\displaystyle{}\sum_{i=1}^{10}3$$\displaystyle{}\sum_{j=0}^{...
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Kenneth Rosen Edition 7 Exercise 2.4 Question 27 (Page No. 169)
Show that if $a_{n}$ denotes the $nth$ positive integer that is not a perfect square, then $a_{n} = n + \{\sqrt{n}\},$ where $\{x\}$ denotes the integer closest to the real number $x.$
Show that if $a_{n}$ denotes the $nth$ positive integer that is not a perfect square, then $a_{n} = n + \{\sqrt{n}\},$ where $\{x\}$ denotes the integer closest to the re...
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