- #1

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- 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

##\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