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Suppose positive real numbers $n_1>n_2>n_3>n_4...$ with these properties are given and you have the sum of $ n_1+n_2+n_3+n_4...$ Is it possible to determine on the basis of this information whether or not the series will converge?

zhw.
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St.Clair Bij
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    No : $\sum\frac{1}{n}$ won't converge whereas $\sum\frac{1}{n^2}$ will. – Balloon Oct 22 '15 at 18:23
  • If it won't converge will it then necessarily diverge to infinity? – St.Clair Bij Oct 22 '15 at 18:25
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    Yes because you are adding positive terms : the case where the limit doesn't exists is excluded. – Balloon Oct 22 '15 at 18:26
  • Thank you for your answers. I don't see how adding positive terms will necessarily lead to infinity: since $\sum \frac{1}{n^2}$ is also positive but does not lead to infinity. – St.Clair Bij Oct 22 '15 at 18:31
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    I didn't say that ; I just said there is two possibilities, convergence to a positive term or divergence (the case $\sum(-1)^n$ where the limit doesn't exists can't be encounter if you considerate positive numbers). – Balloon Oct 22 '15 at 18:38
  • I guess I need some help understanding that. if I understood correctly $\sum \frac {1} {n}$ will diverge, and will go toward infinity. The part which I don't understand is why does it go toward infinity? (And will not stay under an arbitrary number like 1 million or so). – St.Clair Bij Oct 22 '15 at 18:41
  • You can show that the partial sum $\sum_{k=0}^n\frac{1}{k}$ is equivalent to $\ln(n)$. Then it will grow slowly, but surely to $+\infty$. – Balloon Oct 22 '15 at 18:44
  • Let us continue this discussion in chat. – St.Clair Bij Oct 22 '15 at 18:45
  • @St.ClairBij There have been multiple questions on why the harmonic series diverges. See for example here. – Eff Oct 22 '15 at 19:16

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No, the series might converge or diverge. The two classic examples are the harmonic series, $\sum\limits_{n=0}^\infty {\frac{1}{n}}$, which diverges, and the series $\sum\limits_{n=0}^\infty {\frac{1}{n^2}}$, which converges to $\pi^2/6$.

Jacob Brazeal
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