Series Convergence: What Can the Nth Term Test Tell Us?

[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

 

[fresh_42]

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

 

[woopydalan]

I see, 1/sqrt(5)

 

[Mark44]

I see, 1/sqrt(5)

Ok, so what does the nth Term Test for Divergence tell you then?

 

[Mark44]

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.

 

[woopydalan]

Ok, so what does the nth Term Test for Divergence tell you then?

Diverges if it's not 0

 

[Mark44]

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.