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Find-the-supremum-and-infimum-of-each-of-the-following-sequence-a-n-1-2n-b-n-n-2n-1-c-1-n-3-d-sin-npi-2-e-1-n-sin-npi-2-f-1-1-2n-cos-

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Question Number 192738 by Mastermind last updated on 25/May/23
Find the supremum and infimum of each of the following sequence a) {((n−1)/(2n))} b) {(((−)^n n)/(2n+1))} c){((1+(−)^n )/3)} d) {sin((nπ)/2)} e) {(1/n) − sin((nπ)/2)} f) {(1+(1/(2n)))cos((nπ)/3)} Help!
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{supremum}\:\mathrm{and}\:\mathrm{infimum} \\ $$$$\mathrm{of}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{sequence} \\ $$$$ \\ $$$$\left.\mathrm{a}\left.\right)\left.\:\left\{\frac{\mathrm{n}−\mathrm{1}}{\mathrm{2n}}\right\}\:\:\:\:\mathrm{b}\right)\:\left\{\frac{\left(−\right)^{\mathrm{n}} \mathrm{n}}{\mathrm{2n}+\mathrm{1}}\right\}\:\:\:\:\mathrm{c}\right)\left\{\frac{\mathrm{1}+\left(−\right)^{\mathrm{n}} }{\mathrm{3}}\right\} \\ $$$$ \\ $$$$\left.\mathrm{d}\left.\right)\:\left\{\mathrm{sin}\frac{\mathrm{n}\pi}{\mathrm{2}}\right\}\:\:\:\:\mathrm{e}\right)\:\left\{\frac{\mathrm{1}}{\mathrm{n}}\:−\:\mathrm{sin}\frac{\mathrm{n}\pi}{\mathrm{2}}\right\} \\ $$$$ \\ $$$$\left.\mathrm{f}\right)\:\left\{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2n}}\right)\mathrm{cos}\frac{\mathrm{n}\pi}{\mathrm{3}}\right\} \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$
Answered by Rajpurohith last updated on 26/May/23
a)1 b)sup=(1/2) and inf=−(1/2) c)sup=(2/3) and inf=(1/3) d)sup=1 and inf=−1 e)sup=2 and inf=0 f)sup=(3/2) and inf=1
$$\left.{a}\left.\right)\mathrm{1}\:\:\:\:\:{b}\right){sup}=\frac{\mathrm{1}}{\mathrm{2}}\:{and}\:{inf}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.{c}\right){sup}=\frac{\mathrm{2}}{\mathrm{3}}\:\:{and}\:{inf}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\left.{d}\right){sup}=\mathrm{1}\:{and}\:{inf}=−\mathrm{1} \\ $$$$\left.{e}\right){sup}=\mathrm{2}\:{and}\:{inf}=\mathrm{0} \\ $$$$\left.{f}\right){sup}=\frac{\mathrm{3}}{\mathrm{2}}\:{and}\:{inf}=\mathrm{1} \\ $$
Commented by Mastermind last updated on 26/May/23
Thank you but i need solution how you got each
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{but}\:\mathrm{i}\:\mathrm{need}\:\mathrm{solution} \\ $$$$\mathrm{how}\:\mathrm{you}\:\mathrm{got}\:\mathrm{each} \\ $$
Commented by MM42 last updated on 27/May/23
sir your answers are incorrect.please pay atyention my answers
$${sir} \\ $$$${your}\:{answers}\:{are}\:{incorrect}.{please}\:{pay}\:{atyention} \\ $$$${my}\:{answers} \\ $$
Answered by MM42 last updated on 27/May/23
a)0,(1/4),(1/3),(3/8),...→(1/2) ⇒inf=0 & sup=(1/2) b) for the “n=odd ” −(1/3),−(3/7),−(5/(11)),...→inf=−(1/2) & sup=−(1/3) (i) for the “n=even” (2/5),(4/9),(6/(13)),...→inf=(2/5) & sup=(1/2) (ii) (i) , (ii) ⇒inf=−(1/2) & sup=(1/2) c) for the “n=odd ” 0,0,0,...→inf=sup=0 for the “n=even ” (2/3),(2/3),(2/3),...→inf=sup=(2/3) ⇒inf=0 & sup=(2/3) d)1,0,−1,0,1,0,−1,...⇒ inf=−1 & sup=1 e) for the “n=even”e_n =(1/n)⇒ (1/2),(1/4),(1/6),...→0⇒inf=0 & sup=(1/2) (i) for the “n=4k+1 ”e_n =(1/n)−1⇒0,− (4/5),−(8/9)...→−1⇒inf=−1 & sup=0 (ii) for the “n=4k+3 ”e_n =(1/n)+1⇒ (4/3),(8/7),((12)/(11))...→1⇒inf=1 & sup=(4/3) (iii) (i),(ii),(iii)⇒inf=−1 & sup=(4/3) f) for the “n=6k ”f_n =1+(1/(2n))⇒((13)/(12)), ((25)/(24)),((37)/(36)),...→1⇒inf=1 & sup=((13)/(12)) (i) for the “n=6k−3 ”f_n =−(1+(1/(2n)))⇒−(7/6), −((19)/(18)),−((31)/(30)),...→−1⇒inf=−(7/6) & sup=−1 (ii) for the “n=1,5,7,11,13,..”f_n =(1/2)(1+(1/(2n)))⇒(3/4), ((11)/(20)),((15)/(28)),...→(1/2)⇒inf=(1/2) & sup=(3/4) (iii) for the “n=2,4,8,10,14,..”f_n =−(1/2)(1+(1/(2n)))⇒−(5/8), −(9/(16)),−((17)/(32)),−((29)/(56))...→−(1/2)⇒inf=−(5/8) & sup=−(1/2) (iv) (i),(ii),(iii),(iv)⇒inf=−(7/6) & sup=((13)/(12))
$$\left.{a}\right)\mathrm{0},\frac{\mathrm{1}}{\mathrm{4}},\frac{\mathrm{1}}{\mathrm{3}},\frac{\mathrm{3}}{\mathrm{8}},…\rightarrow\frac{\mathrm{1}}{\mathrm{2}}\:\:\Rightarrow{inf}=\mathrm{0}\:\:\&\:\:{sup}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.{b}\right) \\ $$$${for}\:{the}\:“{n}={odd}\:''\:−\frac{\mathrm{1}}{\mathrm{3}},−\frac{\mathrm{3}}{\mathrm{7}},−\frac{\mathrm{5}}{\mathrm{11}},…\rightarrow{inf}=−\frac{\mathrm{1}}{\mathrm{2}}\:\&\:\:{sup}=−\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\left({i}\right) \\ $$$${for}\:{the}\:“{n}={even}''\:\frac{\mathrm{2}}{\mathrm{5}},\frac{\mathrm{4}}{\mathrm{9}},\frac{\mathrm{6}}{\mathrm{13}},…\rightarrow{inf}=\frac{\mathrm{2}}{\mathrm{5}}\:\:\&\:\:{sup}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\left({ii}\right) \\ $$$$\left({i}\right)\:,\:\left({ii}\right)\:\Rightarrow{inf}=−\frac{\mathrm{1}}{\mathrm{2}}\:\:\&\:\:{sup}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\: \\ $$$$\left.{c}\right) \\ $$$${for}\:{the}\:“{n}={odd}\:''\:\mathrm{0},\mathrm{0},\mathrm{0},…\rightarrow{inf}={sup}=\mathrm{0} \\ $$$${for}\:{the}\:“{n}={even}\:''\:\frac{\mathrm{2}}{\mathrm{3}},\frac{\mathrm{2}}{\mathrm{3}},\frac{\mathrm{2}}{\mathrm{3}},…\rightarrow{inf}={sup}=\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\Rightarrow{inf}=\mathrm{0}\:\:\&\:\:{sup}=\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\left.{d}\right)\mathrm{1},\mathrm{0},−\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{0},−\mathrm{1},…\Rightarrow\:{inf}=−\mathrm{1}\:\&\:{sup}=\mathrm{1} \\ $$$$\left.{e}\right) \\ $$$${for}\:{the}\:“{n}={even}''{e}_{{n}} =\frac{\mathrm{1}}{{n}}\Rightarrow\:\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{4}},\frac{\mathrm{1}}{\mathrm{6}},…\rightarrow\mathrm{0}\Rightarrow{inf}=\mathrm{0}\:\:\&\:\:{sup}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\left({i}\right) \\ $$$${for}\:{the}\:“{n}=\mathrm{4}{k}+\mathrm{1}\:''{e}_{{n}} =\frac{\mathrm{1}}{{n}}−\mathrm{1}\Rightarrow\mathrm{0},−\:\frac{\mathrm{4}}{\mathrm{5}},−\frac{\mathrm{8}}{\mathrm{9}}…\rightarrow−\mathrm{1}\Rightarrow{inf}=−\mathrm{1}\:\:\&\:\:{sup}=\mathrm{0}\:\:\:\left({ii}\right) \\ $$$${for}\:{the}\:“{n}=\mathrm{4}{k}+\mathrm{3}\:''{e}_{{n}} =\frac{\mathrm{1}}{{n}}+\mathrm{1}\Rightarrow\:\frac{\mathrm{4}}{\mathrm{3}},\frac{\mathrm{8}}{\mathrm{7}},\frac{\mathrm{12}}{\mathrm{11}}…\rightarrow\mathrm{1}\Rightarrow{inf}=\mathrm{1}\:\:\&\:\:{sup}=\frac{\mathrm{4}}{\mathrm{3}}\:\:\:\left({iii}\right) \\ $$$$\left({i}\right),\left({ii}\right),\left({iii}\right)\Rightarrow{inf}=−\mathrm{1}\:\&\:{sup}=\frac{\mathrm{4}}{\mathrm{3}} \\ $$$$\left.{f}\right) \\ $$$${for}\:{the}\:“{n}=\mathrm{6}{k}\:''{f}_{{n}} =\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{n}}\Rightarrow\frac{\mathrm{13}}{\mathrm{12}},\:\frac{\mathrm{25}}{\mathrm{24}},\frac{\mathrm{37}}{\mathrm{36}},…\rightarrow\mathrm{1}\Rightarrow{inf}=\mathrm{1}\:\:\&\:\:{sup}=\frac{\mathrm{13}}{\mathrm{12}}\:\:\:\left({i}\right) \\ $$$${for}\:{the}\:“{n}=\mathrm{6}{k}−\mathrm{3}\:''{f}_{{n}} =−\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{n}}\right)\Rightarrow−\frac{\mathrm{7}}{\mathrm{6}},\:−\frac{\mathrm{19}}{\mathrm{18}},−\frac{\mathrm{31}}{\mathrm{30}},…\rightarrow−\mathrm{1}\Rightarrow{inf}=−\frac{\mathrm{7}}{\mathrm{6}}\:\:\&\:\:{sup}=−\mathrm{1}\:\:\:\left({ii}\right) \\ $$$${for}\:{the}\:“{n}=\mathrm{1},\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},..''{f}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{n}}\right)\Rightarrow\frac{\mathrm{3}}{\mathrm{4}},\:\frac{\mathrm{11}}{\mathrm{20}},\frac{\mathrm{15}}{\mathrm{28}},…\rightarrow\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow{inf}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\&\:\:{sup}=\frac{\mathrm{3}}{\mathrm{4}}\:\:\:\left({iii}\right) \\ $$$${for}\:{the}\:“{n}=\mathrm{2},\mathrm{4},\mathrm{8},\mathrm{10},\mathrm{14},..''{f}_{{n}} =−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{n}}\right)\Rightarrow−\frac{\mathrm{5}}{\mathrm{8}},\:−\frac{\mathrm{9}}{\mathrm{16}},−\frac{\mathrm{17}}{\mathrm{32}},−\frac{\mathrm{29}}{\mathrm{56}}…\rightarrow−\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow{inf}=−\frac{\mathrm{5}}{\mathrm{8}}\:\:\&\:\:{sup}=−\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\left({iv}\right) \\ $$$$\left({i}\right),\left({ii}\right),\left({iii}\right),\left({iv}\right)\Rightarrow{inf}=−\frac{\mathrm{7}}{\mathrm{6}}\:\&\:\:{sup}=\frac{\mathrm{13}}{\mathrm{12}} \\ $$

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