# Determine whether the following series converge or diverge using the properties.

Determine whether the following series converge or diverge using the properties.
$$\displaystyle{\sum_{{{k}={0}}}^{{\infty}}}{\frac{{{10}}}{{{k}^{{{2}}}+{9}}}}$$

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Usamah Prosser

Step 1
Calculation:
$$\displaystyle{\sum_{{{k}={0}}}^{{\infty}}}{\frac{{{10}}}{{{k}^{{{2}}}+{9}}}}$$
$$\displaystyle={10}{\sum_{{{k}={0}}}^{{\infty}}}{\frac{{{1}}}{{{k}^{{{2}}}+{9}}}}$$
Applying series comparison test
$$\displaystyle{k}^{{{2}}}+{9}{>}{k}^{{{2}}}$$
$$\displaystyle{\frac{{{1}}}{{{k}^{{{2}}}+{9}}}}{<}{\frac{{{1}}}{{{k}^{{{2}}}}}}$$
$$\displaystyle{\sum_{{{k}={0}}}^{{\infty}}}{\frac{{{1}}}{{{k}^{{{2}}}}}}\Rightarrow$$ convergent by p series test.
Hence by comparison test
$$\displaystyle{\sum_{{{k}={0}}}^{{\infty}}}{\frac{{{1}}}{{{k}^{{{2}}}+{9}}}}$$ is convergent.
Also $$\displaystyle{\sum_{{{k}={0}}}^{{\infty}}}{\frac{{{10}}}{{{k}^{{{2}}}+{9}}}}$$ is convergent.