# Determine series convergence

• woopydalan
In summary, the conversation discusses different tests for determining the convergence or divergence of a series, specifically the divergence test, ratio test, and integral test. The conversation also touches on the idea of indeterminate forms and the importance of fully evaluating a limit before making a conclusion. Ultimately, the conversation concludes that the Nth Term Test for Divergence indicates that the series diverges if the limit is not equal to 0.

#### woopydalan

Homework Statement
Determine whether each of the following series converges or not.
## \sum_{n=1}^{\infty} \frac {n+3}{\sqrt{5n^2+1}}##
Relevant Equations
Divergence test, ratio test, etc
I'm not sure which test is the best to use, so I just start with a divergence test

##\lim_{n \to \infty} \frac {n+3}{\sqrt{5n^2+1}}##
The +3 and +1 are negligible
##\lim_{n \to \infty} \frac {n}{\sqrt{5n^2}}##

So now I have ##\infty / \infty##. So it's not conclusive. Trying ratio test

##\lim_{n \to \infty} \lvert \frac {n+4}{\sqrt{5(n+1)^2+1}} \cdot \frac {\sqrt{5n^2+1}}{n+3} \rvert##
seems to yield 1, so inconclusive

Integral test
## \int_{1}^{\infty} \frac {x+3}{\sqrt{5x^2+1}} dx ##. I could separate
## \int_{1}^{\infty} \frac {x}{\sqrt{5x^2+1}} dx + \int_{1}^{\infty} \frac {3}{\sqrt{5x^2+1}} dx ##
First part of the sum would be u-sub, not sure if I even know how to do the second part of the sum

Think again about ##\displaystyle{\lim_{n \to \infty}\dfrac{n}{\sqrt{5n^2}}}.##

• Delta2
I see, 1/sqrt(5)

• Delta2
woopydalan said:
I see, 1/sqrt(5)
Ok, so what does the nth Term Test for Divergence tell you then?

woopydalan said:
So now I have ∞/∞.
Which means you haven't gone far enough in evaluating the limit. If you get any of the indeterminate forms, such as ##\frac \infty \infty, \frac 0 0, \infty - \infty,## or a few others, there is still work to do.

Mark44 said:
Ok, so what does the nth Term Test for Divergence tell you then?
Diverges if it's not 0

woopydalan said:
Diverges if it's not 0
What I meant was, what does the Nth Term Test tell you about this series, something I think you have now figured out.