The Convergence Theorem for Power Series There are

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The Convergence Theorem for Power Series There are three possibilities for with respect to

The Convergence Theorem for Power Series There are three possibilities for with 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 Convergence Theorem for Power Series (Continued). 2. The series converges for every 3. The series converges at elsewhere . and diverges

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

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

The Direct Comparison Test Let be a series with no negative terms. (a) converges

The Direct Comparison Test Let be 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 some integer N. for all

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

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

The Ratio Test Let be a series with all positive terms, and with Then,

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

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 the integral diverge. and either both converge or both

The p-Series Test 1. converges if p > 1. 2. diverges if p <

The p-Series Test 1. converges if p > 1. 2. diverges if p < 1. 3. diverges if p = 1. (Harmonic Series)

The Limit Comparison Test (LCT) Suppose that and for all (N a positive integer).

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

The Alternating Series Test (Leibniz’s Theorem) The series converges if all three of the

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

The Alternating Series Estimation Theorem If the alternating series satisfies the conditions of Leibniz’s

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

nth-root Test •

nth-root Test •