1. ## The Integral Test

Do the series diverge or converge? and why?

1.) n=1 to infinity (5/n+1)

2.) n=1 to infinity (-2/n(square root n))

3.)n=2 to infinity (ln n/(square root n))

4.)n=1 to infinity (1+ 1/n)^n

5.)n=1 to infinity (n/n^2+1)

2. Originally Posted by TAG16
Do the series diverge or converge? and why?

1.) n=1 to infinity (5/n+1)

2.) n=1 to infinity (-2/n(square root n))

3.)n=2 to infinity (ln n/(square root n))

4.)n=1 to infinity (1+ 1/n)^n

5.)n=1 to infinity (n/n^2+1)
1. $\sum_{n = 1}^{\infty}{\frac{5}{n + 1}} \geq \int_{1}^{\infty}{\frac{5}{x + 1}\,dx}$

$\int_{1}^{\infty}{\frac{5}{x + 1}\,dx} = \lim_{\varepsilon \to \infty}5\ln{(\varepsilon + 1)} - \ln{(0 + 1)} \to \infty$

This tends to $\infty$, and so the series, which is no less than this integral, also tends to $\infty$. So the series diverges.