Comparison test

The comparison test for divergence of infinite series is analogous to the Squeeze Theorem. It states that if sequences $(𝑎 𝑛) \infty 𝑛 = 1$ and $(𝑏 𝑛) \infty 𝑛 = 1$ obey the relationship

0 \leq 𝑎 𝑛 \leq 𝑏 𝑛

for all $𝑛 \in ℕ$ , then the series obey the relation

0 \leq \infty \sum 𝑛 = 1 𝑎 𝑛 \leq \infty \sum 𝑛 = 0 𝑏 𝑛

and therefore the convergence of the latter series implies the convergence of the former, and the divergence of the former series implies the divergence of the former.

Often a series will be compared with a 𝑝-series.

#state/tidy | #SemBr