Tests for series divergence

Limit comparison test

The limit comparison test takes the limit of the ratio between corresponding terms of two sequences in order to compare the rates at which the sequences go to zero. Given two infinite series and such that and for sufficiently large , we find the positive ratio

which gives us three cases:¹

1. means for large , and therefore the series approach scalar multiples of each other. Therefore, they are either both convergent or both divergent.

2. means becomes much larger than in the long run, and therefore the series is an upper bound on . Hence if converges converges.

3. means the opposite of the above case, and hence if converges converges.

This is advantageous over the similar Comparison test in cases where two sequences approach scalings of each other as .

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