List of Series Tests

The Integral Test states that if f is a continuous, positive, decreasing function on [1,∞) and a_n = f(n), then the series Σ a_n is convergent if and only if the improper integral ∫ [1,∞] f(x) dx is __________.

convergent.

A P-series Σ 1/n^p is __________ if p > 1 and is __________ if p ≤ 1.

convergent; divergent.

The Comparison Test states that if Σ b_n is convergent and a_n < b_n for all n, then Σ a_n is also __________.

convergent.

According to the Limit Comparison Test, if lim (n→∞) (a_n/b_n) = c where c is a finite number and c > 0, then either both series converge or both __________.

diverge.

For an alternating series to converge, it must satisfy the conditions that b_(n+1) < b_n for all n and lim(n→∞) b_n = __________.

A series is called __________ convergent if the series of absolute values is convergent.

absolutely.

The Ratio Test is always used for factorials and _________ and never for algebraic functions.

exponentials.

If in the Ratio Test L < 1, then the series is __________ convergent.

absolutely.

If in the Ratio Test L > 1, then the series is __________.

divergent.

If L = 1 in the Ratio Test, then the test is __________.

inconclusive.

The Root Test states that if lim (n→∞) (n-th root of |a_n|) = L and L < 1, then the series is __________ convergent.

absolutely.

If L = 1 in the Root Test, then the root test is __________.

inconclusive.