If L=lim_(nto oo) (n^(3)(e^(1//n)+e^(2//n)+………+e))/((n+1)^(m)(1^(m)+4^(m)+….+n^(2m))) is non zer finite real, then

If L=lim_(nto oo) (n^(3)(e^(1//n)+e^(2//n)+………+e))/((n+1)^(m)(1^

1.	If L=lim_(nto oo) (n^(3)(e^(1//n)+e^(2//n)+………+e))/((n+1)^(m)(1^(m)+4^(m)+….+n^(2m))) is non zer finite real, then
Answer» `L=3(e-1)` `L=2(e-1)` `m=1//3` `m=1//3` Solution :`L=lim_(n to oo) (n^(3)sum_(r=1)^(n)e^(r//n))/((n+1)^(m)sum_(r=1)^(n)r^(2m))` `=lim_(n to oo) (n^(3)sum_(r=1)^(n)e^(r//n) . 1/n)/((n+1)^(m)n^(2m)sum_(r=1)^(n)(r/n)^(2m) . 1/n)` `=lim_(n to oo) (n^(3))/((n+1)^(m)n^(2m)) . (lim_(nto oo) 1/n sum_(r=1)^(n)e^(r//n))/(lim_(nto oo) 1/n sum_(r=1)^(n)(r/n)^(2m))` `=lim_(nto oo) (n^(3))/((n^(3)+n^(2))m) . (int_(0)^(1)e^(x)dx)/(int_(0)^(1)x^(2m)dx)` For `L` to be non-zero finite `m=1` `:. L=(int_(0)^(1)e^(x)dx)/(int_(0)^(1)x^(2)dx)=(e-1)/(1//3) =3(e-1)`

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If L=lim_(nto oo) (n^(3)(e^(1//n)+e^(2//n)+………+e))/((n+1)^(m)(1^(m)+4^(m)+….+n^(2m))) is non zer finite real, then

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