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QUESTION

Determine whether or not the series
$\ds\sum_{k=0}^\infty\frac{k+1}{k+4}$ is convergent.

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ANSWER

$\ds
u_k=\frac{k+1}{k+4}=\frac{k(1+\frac{1}{k})}{k(1+\frac{4}{k})}=
\frac{1+\frac{1}{k}}{1+\frac{4}{k}}\to\frac{1}{1}=1$ as
$k\to\infty$ \\ The terms tend to 1 therefore the series is not
convergent.




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