Editing Integral test for convergence (section)

===Remark===
If the improper integral is finite, then the proof also gives the [[upper and lower bounds|lower and upper bounds]]

{{NumBlk|:|<math>\int_N^\infty f(x)\,dx\le\sum_{n=N}^\infty f(n)\le f(N)+\int_N^\infty f(x)\,dx</math>|{{EquationRef|1}}}}

for the infinite series.

Note that if the function <math>f(x)</math> is increasing, then the function <math>-f(x)</math> is decreasing and the above theorem applies.

Many textbooks require the function <math>f</math> to be positive,<ref>{{cite book |last1=Stewart |first1=James |last2=Clegg |first2=Daniel |last3=Watson |first3=Saleem |title=Calculus: Metric Version |date=2021 |publisher=Cengage |isbn=9780357113462 |edition=9}}</ref><ref>{{cite book |last1=Wade |first1=William |title=An Introduction to Analysis |date=2004 |publisher=Pearson Education |isbn=9780131246836 |edition=3}}</ref><ref>{{cite book |last1=Thomas |first1=George |last2=Hass |first2=Joel |last3=Heil |first3=Christopher |last4=Weir |first4=Maurice |last5=Zuleta |first5=José Luis |title=Thomas' Calculus: Early Transcendentals |date=2018 |publisher=Pearson Education |isbn=9781292253114 |edition=14}}</ref> but this condition is not really necessary, since when <math>f</math> is negative and decreasing both <math>\sum_{n=N}^\infty f(n)</math> and <math>\int_N^\infty f(x)\,dx</math> diverge.<ref>{{cite web |url=https://math.stackexchange.com/q/3577379 |title=Why does it have to be positive and decreasing to apply the integral test? |author=savemycalculus |website=Mathematics Stack Exchange |access-date=2020-03-11}}</ref>{{better source needed|date=August 2024}}