# Is a positive series convergent if the terms decrease?

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?

• 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
• 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
• 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

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 $\pi^2/6$.