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Let $\left\{a_{n}\right\}$ be a strictly increasing sequence of positive numbers,

Do $$\sum_{n=0}^{\infty}\frac{1}{a_{n}}$$ and $$\sum_{n=0}^{\infty}\frac{a_{n-1}}{a_{n}}$$ and $$\sum_{n=0}^{\infty}e^{-a_{n}}$$ MUST converge ? I have tried ratio test,but it seems does not work. How to test the series converge or not?

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take $a_n=n+1$, then $(1)$ does not converge. Do you have additional informations? – Jean-Sébastien Oct 27 '12 at 17:50
For each series, try finding a sequence $a_n$ so that it doesn't converge. Remember, you only need a single counterexample for each one. – Grumpy Parsnip Oct 27 '12 at 17:50
up vote 4 down vote accepted

Not one of these necessarily converges.

For the first one, consider the harmonic series, where the $a_n$ are $1, 2, 3, \ldots$

For the second one, rig it up so the $n$th term doesn't goes to $0$, e.g., $a_n$ are $1, 1.1, 1.11, 1.111, \ldots$

For the third one, pull a similar trick, e.g., $a_n$ are $0.9, 0.99, 0.999, \ldots$.

In each case, the $a_n$'s are strictly increasing sequences of positive numbers, but the corresponding series diverge to infinity. The problem is that these series are too general as stated; you'll need more specificity to guarantee convergence.

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