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Math Help - The Integral Test

  1. #1
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    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)
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  2. #2
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    Quote Originally Posted by TAG16 View Post
    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.
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