Series (mathematics) - Convergence Tests

Convergence Tests

• n-th term test: If lim_n→∞ a_n ≠ 0 then the series diverges.
• Comparison test 1: If ∑b_n is an absolutely convergent series such that |a_n | ≤ C |b_n | for some number C  and for sufficiently large n, then ∑a_n  converges absolutely as well. If ∑|b_n | diverges, and |a_n | ≥ |b_n | for all sufficiently large n, then ∑a_n  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
• Comparison test 2: If ∑b_n  is an absolutely convergent series such that |a_n+1 /a_n | ≤ |b_n+1 /b_n | for sufficiently large n, then ∑a_n  converges absolutely as well. If ∑|b_n | diverges, and |a_n+1 /a_n | ≥ |b_n+1 /b_n | for all sufficiently large n, then ∑a_n  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n  alternate in sign).
• Ratio test: If there exists a constant C < 1 such that |a_n+1/a_n|<C for all sufficiently large n, then ∑a_n converges absolutely. When the ratio is less than 1, but not less than a constant less than 1, convergence is possible but this test does not establish it.
• Root test: If there exists a constant C < 1 such that |a_n|1/n ≤ C for all sufficiently large n, then ∑a_n converges absolutely.
• Integral test: if ƒ(x) is a positive monotone decreasing function defined on the interval [1, ∞) with ƒ(n) = a_n for all n, then ∑a_n converges if and only if the integral  ∫₁∞ ƒ(x) dx is finite.
• Cauchy's condensation test: If a_n is non-negative and non-increasing, then the two series  ∑a_n  and  ∑2ka_(2k) are of the same nature: both convergent, or both divergent.
• Alternating series test: A series of the form ∑(−1)n a_n (with a_n ≥ 0) is called alternating. Such a series converges if the sequence a_n is monotone decreasing and converges to 0. The converse is in general not true.
• For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.