October 30th, 2016, 08:09 AM  #1 
Member Joined: Aug 2016 From: illinois Posts: 46 Thanks: 0  Series Convergence
If you are using the integral test to find if a series is convergent or divergent, and find the integral converges to a negative number, does that mean the series still converges.

October 30th, 2016, 08:24 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra 
The integral test operates on positive, monotonic functions only (from memory). You should investigate the integral of $f(x)$ to see if the series is absolutely convergent.
Last edited by v8archie; October 30th, 2016 at 08:30 AM. 
October 31st, 2016, 08:21 AM  #3 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Integral Test
Let f(x) be positive, continuous , and nonincreasing for x$\displaystyle \geq$ N and a$\displaystyle _{n}$ =f(n) for n$\displaystyle \geq$N. Then a$\displaystyle _{n}$ converges or diverges according as $\displaystyle \int_{N}^{\infty}$f(x)dx converges or diverges. 

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convergence, series 
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