The Convergence Theorem for Power Series There are three possibilities forwith respect to convergence: 1.There is a positive number R such that the series diverges for but converges for. The series may or may not converge at either of the endpoints and

The Convergence Theorem for Power Series (Continued). 2. The series converges for every. 3. The series converges at and diverges elsewhere

The n th-Term Test for Divergence diverges if fails to exist or is different from zero.

The Direct Comparison Test Letbe a series with no negative terms. (a) converges if there is a convergent series with for all for some integer N. (b) diverges if there is a divergent series of nonnegative terms with for all for some integer N.

Absolute Convergence Absolute Convergence Implies Convergence If the series of absolute value converges, then converges absolutely. If converges, then converges.

The Ratio Test Let be a series with all positive terms, and with Then, (a) The series converges if L < 1, (b) The series diverges if L > 1, (c) The test is inconclusive if L = 1.

The Integral Test Let be a sequence of positive terms. Suppose that, where f is a continuous, positive, decreasing function of x for all ( N a positive integer). Then the series and the integral either both converge or both diverge.

The p- Series Test 1. converges if p > diverges if p < diverges if p = 1. (Harmonic Series)

The Limit Comparison Test (LCT) 1. If then and both converge or both diverge. Suppose that and for all ( N a positive integer). 2.If and converges, then converges. 3.If and diverges, then diverges.

The Alternating Series Test (Leibniz’s Theorem) The series converges if all three of the following conditions are satisfied: 1. each is positive. 2. for all, for some integer N. 3.

The Alternating Series Estimation Theorem If the alternating series satisfies the conditions of Leibniz’s Theorem, then the truncation error for the n th partial sum is less than and has the same sign as the first unused term.

nth-root Test