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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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    $(a^n+b^n)^{\frac 1n}\leq 2^{\frac 1n}b$ so for $n$ large enough, $a^n+b^n\leq \left(b+\frac{1-b}2\right)^n=\left(\frac{b+1}2\right)^n$. But I don't know how do we have to use the root test, since $0\leq a^n+b^n\leq 2b^n$, and the convergence of $\sum_n b^n$ is know (in fact, the root test is made to compare the series with a geometric series).2012-01-14
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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.