Question

For each ninN and xin[0,infty ), let f_(n)(x)=(x^(n+1))/(x^(n)+2^(n)). (a) Find the pointwise limit f(x) of the sequence f_(n)(x) on [0,infty ). [6 marks] (b) Argue that the convergence f_(n)->f is not uniform on [0,infty ). [5 marks] (c) Is f(x) integrable on 0,2022 ? Is there a differentiable function F(x) on [0,infty ) such that F^(')(x)=f(x) ? Justify your answers. [6 marks] (d) Show that the convergence f_(n)->f is uniform on [b,infty ) for any b>2. [8 marks] xn+1 4. For each n E N and x E[0,oo),let fn(x) = xn+2n (a) Find the pointwise limit f(x) of the sequence fn(x) on [0,oo) [6 marks] (b) Argue that the convergence fn - f is not uniform on [0,oo) [5 marks] (c) Is f(x) integrable on [0,2022]? Is there a differentiable function F(c) on 0,oo) such that F'(x) = f()? Justify your answers. [6 marks] (d) Show that the convergence fn -> f is uniform on [b,oo) for any b > 2 [8 marks]

          For each ninN and xin[0,infty ), let f_(n)(x)=(x^(n+1))/(x^(n)+2^(n)).
(a) Find the pointwise limit f(x) of the sequence f_(n)(x) on [0,infty ).
[6 marks]
(b) Argue that the convergence f_(n)->f is not uniform on [0,infty ).
[5 marks]
(c) Is f(x) integrable on 0,2022 ? Is there a differentiable function F(x) on
[0,infty ) such that F^(')(x)=f(x) ? Justify your answers.
[6 marks]
(d) Show that the convergence f_(n)->f is uniform on [b,infty ) for any b>2.
[8 marks]
xn+1 4. For each n E N and x E[0,oo),let fn(x) = xn+2n
(a) Find the pointwise limit f(x) of the sequence fn(x) on [0,oo)
[6 marks]
(b) Argue that the convergence fn - f is not uniform on [0,oo)
[5 marks]
(c) Is f(x) integrable on [0,2022]? Is there a differentiable function F(c) on 0,oo) such that F'(x) = f()? Justify your answers.
[6 marks]
(d) Show that the convergence fn -> f is uniform on [b,oo) for any b > 2
[8 marks]

for each ninn and xin0infty let fnxxn1xn2n a find the pointwise limit fx of the sequence fnx on 0infty 6 marks b argue that the convergence fn f is not uniform on 0infty 5 marks c is fx i 13206

Added by Christopher H.

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