Nth-term test

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In mathematics, the nth-term test for divergence^[1] is a simple test for the divergence of an infinite series:

If ${\displaystyle \lim _{n\to \infty }a_{n}\neq 0}$ or if the limit does not exist, then ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ diverges.

Many authors do not name this test or give it a shorter name.^[2]

When testing if a series converges or diverges, this test is often checked first due to its ease of use.

In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-Archimedean ultrametric triangle inequality.