MATH 21
Convergence Tests

Test for Divergence If lim_n®¥ a_n ¹ 0, then åa_n diverges.

Tests for Positive Series

Integral Test If f(x) is a positive, continuous, decreasing function on [1,¥) and a_n = f(n) then the series åa_n converges if and only if the integral ò₁^¥ f(x)dx converges.

Comparison Test Suppose 0 £ a_n £ b_n for all n sufficiently large.

If åb_n converges, then åa_n converges.

If åa_n diverges, then åb_n diverges.

Limit Comparison Test Suppose 0 £ a_n, b_n and that

lim n ®¥  an bn = L.

If L = 0, then if åb_n converges, then åa_n converges and, if åa_n diverges, then åb_n diverges.

If 0 < L < ¥ then åa_n converges if and only if åb_n converges.

If L = ¥, then if åa_n converges, then åb_n converges and, if åb_n diverges, then åa_n diverges.

Test for Series with Positive and Negative terms

Alternating Series Test If {b_n} is a decreasing sequence whose limit is 0, then the series

å (-1)n bn     and     å (-1)n+1 bn

both converge.

Absolute Convergence Test If å|a_n| converges, then åa_n converges.

Ratio Test Suppose

lim n®¥  |an+1| |an| = R.

If R < 1, then åa_n is absolutely convergent.

If R > 1, then åa_n is divergent.

If R = 1, then the test is inconclusive.

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