# Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$f(x)=\sum_{n\geq 0}a_n x^n$$ converges absolutely for all $x$. Under what conditions on $\{a_n\}$ will we have $f(x)\geq 0$ for all $x\leq 0$?

• may the sequence $\{a_n\}_{n=0}^{\infty}$ should be decreasing and converge – zeraoulia rafik Jun 28 '15 at 22:14
• The sequence $\{a_n\}_{n=0}^{\infty}$ should be decreasing and converge vers $0$ as :lim$suplog (a_{n})=-\infty$ , $n \to \infty$ hence you can use leibnez theorem for (Alternating series) – zeraoulia rafik Jun 28 '15 at 22:29