I'd like a hint for the question:
Thanks.
I'd like a hint for the question:
Thanks.
Put $c_n = a^n + b^n$; then $$c_n = b^n\left(1 + {a^n\over b^n}\right);$$ as $n\to\infty$ $$\root{n}\of{|c_n|} = b\left(1 + {a^n\over b^n}\right)^{1/n} \rightarrow b. $$ The root test is conclusive since $|b| < 1$.
Now for the ratio test. As $n\to\infty$, $${c_{n+1}\over c_n} = b\left(1 + {a^n\over b^n}\right)^{-1}\left(1 + {a^{n+1}\over b^{n+1}}\right) \rightarrow b.$$
Both tests are conclusive and yield the same result.