P-series

s_n = converges if p > 1 (just like converges when p > 1)

Ratio

A series converges if the ratio lim < 1.

A series diverges if the ratio lim > 1.

A series either converges or diverges if the ratio lim = 1.

(inconclusive)

ex: If s_n = , then a_n =

The ratio test says that = = à < 1, so s_n converges.

Note: s_n is geometric with a₁ = and r = .

So s_n = = = , so s_n converges to .

Root

A series s_n converges if < 1 (and diverges if the root is greater than 1, and is inconclusive if the root is equal to one, just like the ratio test)

Ex: s_n = , so a_n =

The root test says that = à < 1 so s_n converges.

Note: lim = 1 (Use logarithms to prove this!)

Ex: s_n = so a_n =

Then the root test says = = -> 0 so the series converges.

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