convergetr or divergent? sum_(n=1)^(oo)(1)/(n^(2)+4)

Identify type of series: Identify the type of series. The given series is an infinite series of the form sum_(n=1)^(oo)(1)/(n^(2)+4). This is a series where the terms are given by a rational function of n.Apply convergence test: Apply a convergence test. To determine if the series converges or diverges, we can use the comparison test. We will compare our series to the p-series sum_(n=1)^(oo)(1)/(n^p), which is known to converge if p > 1 and diverge if p ≤ 1.Compare to known series: Compare to a known convergent p-series. The terms (1)/(n^(2)+4) are always less than or equal to (1)/(n^2) for n ≥ 1. Since the p-series sum_(n=1)^(oo)(1)/(n^2) converges (because p = 2 > 1), our series will also converge by the comparison test.Conclude convergence: Conclude the convergence of the series. Since the terms of our series are smaller than the corresponding terms of a convergent p-series, and the comparison test confirms that if a series with smaller positive terms converges, then our original series also converges.State final answer: State the final answer. The series sum_(n=1)^(oo)(1)/(n^(2)+4) is convergent.

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convergent or divergent? $\newline$ $\sum_{n=1}^{\infty}\frac{1}{n^2+4}$

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Q. convergent or divergent?

\newline

\sum_{n=1}^{\infty}\frac{1}{n^2+4}

Identify type of series: Identify the type of series. The given series is an infinite series of the form $\sum_{n=1}^{\infty}\frac{1}{n^{2}+4}$ . This is a series where the terms are given by a rational function of $n$ .
Apply convergence test: Apply a convergence test. $\newline$ To determine if the series converges or diverges, we can use the comparison test. We will compare our series to the p-series $\sum_{n=1}^{\infty}\frac{1}{n^p}$ , which is known to converge if p > 1 and diverge if $p \leq 1$ .
Compare to known series: Compare to a known convergent p-series. $\newline$ The terms $(1)/(n^{2}+4)$ are always less than or equal to $(1)/(n^2)$ for $n \geq 1$ . Since the p-series $\sum_{n=1}^{\infty}(1)/(n^2)$ converges (because p = 2 > 1), our series will also converge by the comparison test.
Conclude convergence: Conclude the convergence of the series. $\newline$ Since the terms of our series are smaller than the corresponding terms of a convergent $p$ -series, and the comparison test confirms that if a series with smaller positive terms converges, then our original series also converges.
State final answer: State the final answer. $\newline$ The series $\sum_{n=1}^{\infty}\frac{1}{n^{2}+4}$ is convergent.