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Question

# Test the series for convergence.$\sum_{k=1}^{\infty} \frac{1}{\sqrt{2 k+3}}$

Solution

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Integral Test: Let $a_n = f (n)$, where f is a positive, decreasing, and continuous function of $x$ for $x \geq 1$

(i) If $\displaystyle \int ^{ \infty}_{1} f(x) dx$ converges then $\sum\limits_{n=1}^{\infty} a_n$ converges

(ii) If $\displaystyle \int ^{ \infty}_{1} f(x) dx$ diverges then $\sum\limits_{n=1}^{\infty} a_n$ diverges

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