`sum_(n=1)^oo 1/(2n+3)^3` Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.

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For the integral test, if f is positive, continuous and decreasing for `x>=1` and `a_n=f(n)` , then `sum_(n=1)^ooa_n` and `int_1^oof(x)dx` either both converge or diverge.

Now, `f(x)=1/(2x+3)^3`

Now function is positive and continuous.

Let's determine whether f(x) is decreasing, by finding its derivative `f'(x)`





`f'(x)<0` , so the function is decreasing

Because f(x) satisfies the conditions for the integral test, we can apply integral test.








So f(x) converges.

Therefore, `sum_(n=1)^oo1/(2n+3)^2` converges.

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