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

Maiclubk 2021-08-17 Answered
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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Expert Answer

Usamah Prosser
Answered 2021-08-18 Author has 7859 answers

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.

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