Answer to Question #249180 in Real Analysis for Raghad

Question

Show that if Zn = (a + b")

Answer 1

if $z_n=(a^n+b^n)^{1/n}$ where $0<a<b$ , then

$\displaystyle{\lim_{n\to \infin}}z_n=b$

$(a^n+b^n)^{1/n}=b((a/b)^n+1)^{1/n}$

$\displaystyle{\lim_{n\to \infin}}(a/b)^n=0$

$\displaystyle{\lim_{n\to \infin}}((a/b)^n+1)^{1/n}=0$

$\displaystyle{\lim_{n\to \infin}}(b((a/b)^n+1)^{1/n})=b\cdot1=b$