Absolute Convergence and Ratio and Root Tests

Ratio Test

In this example, the ratio of adjacent terms in the blue sequence converges to $L=\frac{1}{2}$. We choose $r = \frac{L+1}{2} = \frac{3}{4}$. Then the blue sequence is dominated by the red sequence for all $n \geq 2$. The red sequence converges, so the blue sequence does as well.

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Absolute Convergence and Ratio and Root Tests

Ratio Test

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