Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 74109 by FCB last updated on 19/Nov/19

Commented by mr W last updated on 19/Nov/19

for a_k >0 and Σ_(k=1) ^n a_k =1 when a_k =(1/n), max=(n−1)^n

$${for}\:{a}_{{k}} >\mathrm{0}\:{and}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{k}} =\mathrm{1} \\ $$$${when}\:{a}_{{k}} =\frac{\mathrm{1}}{{n}}, \\ $$$${max}=\left({n}−\mathrm{1}\right)^{{n}} \\ $$

Commented by MJS last updated on 20/Nov/19

((1/a_n )−1)Π_(k=1) ^(n−1) ((1/a_k )−1)≈ [let a_n very close to 0 and a_1 =a_2 =...=a_(n−1) ] ≈((1/a_n )−1)((1/(1/(n−1)))−1)^(n−1) =((1/a_n )−1)(n−2)^(n−1) n>2: a_n →0^+ ⇒ ((1/a_n )−1) → +∞ the same Sir Aifour found before

$$\left(\frac{\mathrm{1}}{{a}_{{n}} }−\mathrm{1}\right)\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}\left(\frac{\mathrm{1}}{{a}_{{k}} }−\mathrm{1}\right)\approx \\ $$$$\:\:\:\:\:\left[\mathrm{let}\:{a}_{{n}} \:\mathrm{very}\:\mathrm{close}\:\mathrm{to}\:\mathrm{0}\:\mathrm{and}\:{a}_{\mathrm{1}} ={a}_{\mathrm{2}} =...={a}_{{n}−\mathrm{1}} \right] \\ $$$$\approx\left(\frac{\mathrm{1}}{{a}_{{n}} }−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\frac{\mathrm{1}}{{n}−\mathrm{1}}}−\mathrm{1}\right)^{{n}−\mathrm{1}} =\left(\frac{\mathrm{1}}{{a}_{{n}} }−\mathrm{1}\right)\left({n}−\mathrm{2}\right)^{{n}−\mathrm{1}} \\ $$$${n}>\mathrm{2}: \\ $$$${a}_{{n}} \rightarrow\mathrm{0}^{+} \:\Rightarrow\:\left(\frac{\mathrm{1}}{{a}_{{n}} }−\mathrm{1}\right)\:\rightarrow\:+\infty \\ $$$$ \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{Sir}\:\mathrm{Aifour}\:\mathrm{found}\:\mathrm{before} \\ $$

Commented by FCB last updated on 20/Nov/19

thanks

$$\mathrm{thanks} \\ $$

Answered by ajfour last updated on 19/Nov/19

a_1 →0 , a_2 , a_3 ,...,a_n → (1/(n−1)) ⇒ (→∞)×(n−2)^(n−1) →∞ .

$${a}_{\mathrm{1}} \rightarrow\mathrm{0}\:,\:\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,...,{a}_{{n}} \rightarrow\:\frac{\mathrm{1}}{{n}−\mathrm{1}}\: \\ $$$$\Rightarrow\:\left(\rightarrow\infty\right)×\left({n}−\mathrm{2}\right)^{{n}−\mathrm{1}} \:\rightarrow\infty\:. \\ $$

Commented by FCB last updated on 19/Nov/19

prove that sir

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{sir} \\ $$

Commented by ajfour last updated on 19/Nov/19

n should be finite>2 and a_1 →0 can be possible as well!

$${n}\:{should}\:{be}\:{finite}>\mathrm{2}\:\:{and}\:{a}_{\mathrm{1}} \rightarrow\mathrm{0} \\ $$$${can}\:{be}\:{possible}\:{as}\:{well}! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com