Criterio Integrale Enunciato Sia a0∈Na_0 \in \mathbb{N}a0∈N e f:[n0,+∞]→Rf:[n_0, +\infty]\to \mathbb{R}f:[n0,+∞]→R una funzione tale che: f≥0, f decrescente in D,limn→+∞f(x)=0 f\geq 0,\ f\ decrescente\ in\ D, \lim_{n\to +\infty} f(x) = 0 f≥0, f decrescente in D,n→+∞limf(x)=0 Poniamo an=f(n)a_n = f(n)an=f(n): ∑n=n0+∞an conv⇐⇒∫n0+∞f(x)dxconv \sum_{n=n_0}^{+\infty}a_n\ conv \Leftarrow \Rightarrow \int_{n_0}^{+\infty}f(x)dx conv n=n0∑+∞an conv⇐⇒∫n0+∞f(x)dxconv ∑n=n0+∞an div⇐⇒∫n0+∞f(x)dxdiv \sum_{n=n_0}^{+\infty}a_n\ div \Leftarrow \Rightarrow \int_{n_0}^{+\infty}f(x)dx div n=n0∑+∞an div⇐⇒∫n0+∞f(x)dxdiv