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Prove $$\sum^{\infty}_{n=1} \frac{a_{n}-a_{n-1}}{a_{n}}=\infty$$ Where $a_{n}$ is an increasing sequence of positive terms that goes to infinity.

Prove $\sum^{\infty}_{n=1} \frac{a_{n}-a_{n-1}}{a_{n}}=\infty$

In the solution given, it picks an increasing sequence $N_i$ so that $a_{N_{i+1}} > 2 A_{N_{i+1}}$. Why we can do that?

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user10024395
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1 Answers1

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I think it should be $a_{N_{i+1}}>2a_{N_i}$, instead of the upper case one.

That is because $a_i$ is going to infinity, so we can find $a_i$s arbitrarily large, larger than $2a_{N_i}$, and we set that $i$ to be $N_{i+1}$.

Colliot
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