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I'd like a hint for the question:

Let $0, then the serie $ \sum_n( a^n + b^n)$ converges, use the root test to prove that, and show that the ratio test is inconclusive in this case.

Thanks.

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    The ratio test is conclusive in this case...2012-01-14

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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.