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f-n-x-1-n-1-1-x-2-1-x-n-1-2x-1-nx-n-N-n-gt-1-lim-n-f-n-x-n-gt-1-f-x-0-x-




Question Number 4750 by 123456 last updated on 04/Mar/16
f_n (x)= { ((1    n=1)),(((((1−x^2 )...(1−x^n ))/((1−2x)...(1−nx)))    n∈N,n>1)) :}  lim_(n→∞) f_n (x)=?  n>1,f(x)=0,x=?
$${f}_{{n}} \left({x}\right)=\begin{cases}{\mathrm{1}\:\:\:\:{n}=\mathrm{1}}\\{\frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)…\left(\mathrm{1}−{x}^{{n}} \right)}{\left(\mathrm{1}−\mathrm{2}{x}\right)…\left(\mathrm{1}−{nx}\right)}\:\:\:\:{n}\in\mathbb{N},{n}>\mathrm{1}}\end{cases} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{f}_{{n}} \left({x}\right)=? \\ $$$${n}>\mathrm{1},{f}\left({x}\right)=\mathrm{0},{x}=? \\ $$
Commented by prakash jain last updated on 06/Mar/16
First part lim_(n→∞) f_n (x) to be solved later  x=±1 if f(x)=0 for n>1
$$\mathrm{First}\:\mathrm{part}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}{f}_{{n}} \left({x}\right)\:\mathrm{to}\:\mathrm{be}\:\mathrm{solved}\:\mathrm{later} \\ $$$${x}=\pm\mathrm{1}\:{if}\:{f}\left({x}\right)=\mathrm{0}\:\mathrm{for}\:{n}>\mathrm{1} \\ $$

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