comparison test

The series

\\sum_{i=0}^{\\infty}a_{i}

with real $a_{i}$ is absolutely convergent if there is a sequence $(b_{n})_{n\\in\\mathbb{N}}$ with positive real $b_{n}$ such that

\\sum_{i=0}^{\\infty}b_{i}

is and for all sufficiently large $k$ holds $|a_{k}|\\leq b_{k}$ .

Also, the series $\\sum a_{i}$ is divergent if there is a sequence $(b_{n})$ with positive real $b_{n}$ , so that $\\sum b_{i}$ is divergent and $a_{k}\\geq b_{k}$ for all sufficiently large $k$ .

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