The Divergence Test
Also called the nth term test. It never proves convergence — it only ever catches divergence, by checking the one thing every convergent series must do: its terms have to shrink to zero.
If the limit does equal zero, the test is silent — the series could still converge or diverge, and you'd need another test to find out.
Worked Example
Does this series converge or diverge? Apply the Divergence Test to justify your answer.
The Ratio Test
Compares each term to the one before it. If consecutive terms keep shrinking by roughly the same factor, that factor tells you everything.
It's most useful when a term involves an n-th power or a factorial, since the ratio cancels most of the expression away.
Worked Example
Does this series converge or diverge? Apply the Ratio Test to justify your answer.
The Root Test
Compares the size of a term to itself, n times over, by taking the n-th root. It's the natural choice when the whole term is already raised to the power n.
This is Cauchy's test — a close cousin of the Ratio Test that is often stronger, though usually harder to apply by hand.
Worked Example
Does this series converge or diverge? Apply the Root Test to justify your answer.
The Comparison Test
Sits a series next to one you already understand. If your series is always the smaller of the two and the bigger one converges, yours has no choice but to converge too — and vice versa for divergence.
The skill is choosing a sensible bn — usually a p-series or geometric series whose behaviour you already know.
Worked Example
Does this series converge or diverge? Find a suitable series to compare it with.
The Limit Comparison Test
A more forgiving cousin of the Comparison Test. Instead of proving a term-by-term inequality, you only need the two series to behave similarly in the long run.
Ideal for rational functions of n: pick bn to be the dominant term as n → ∞, and the messy lower-order parts fall away in the limit.
Worked Example
Does this series converge or diverge? Find a series to compare it with in the limit.
The Integral Test
Treats the series as a staircase sitting alongside the smooth curve of a related function, and asks whether the area under that curve is finite.
Note this only tells you whether the series converges, not what it converges to — the sum and the integral are rarely equal.
Worked Example
Does this series converge or diverge? Apply the Integral Test to justify your answer.
The Alternating Series Test
Also known as the Leibniz Test. When the signs of the terms flip back and forth, each new term nudges the partial sum past the previous one — so as long as the steps keep shrinking to zero, the sum has to settle somewhere.
Like the Divergence Test, this one only ever points one way — it can prove convergence, but it can never prove divergence.
Worked Example
Does this series converge? Check the conditions of the Alternating Series Test.
The Absolute Convergence Test
Strips the signs away and asks a stronger question: does the series still converge if every term is made positive? If so, the original series was never in any danger.
Absolute convergence is the stronger property: it implies convergence, but not the other way round — a series can converge only conditionally, the way the alternating harmonic series does.
Worked Example
Is this series absolutely convergent, conditionally convergent, or divergent?