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?
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1$\begingroup$ No : $\sum\frac{1}{n}$ won't converge whereas $\sum\frac{1}{n^2}$ will. $\endgroup$ – Balloon Oct 22 '15 at 18:23
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$\begingroup$ If it won't converge will it then necessarily diverge to infinity? $\endgroup$ – St.Clair Bij Oct 22 '15 at 18:25
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1$\begingroup$ Yes because you are adding positive terms : the case where the limit doesn't exists is excluded. $\endgroup$ – Balloon Oct 22 '15 at 18:26
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$\begingroup$ 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. $\endgroup$ – St.Clair Bij Oct 22 '15 at 18:31
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2$\begingroup$ 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). $\endgroup$ – Balloon Oct 22 '15 at 18:38
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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 $\pi^2/6$.
