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[14] EXPAND in powers of x. f(x) = 9x2 sin (9x3) OPTIONAL (bonus): Find the sum of the series. Show work. 6 k 2+4k + 3e = lim ( 1+ * ) p- series K -700 M KF converges psi . = lim ( 1+ + ) diverges . p 4 1 K-700 Review of Convergence Tests, Presented In a Suggested Order for Checking I. Divergence test Compute lim a,. If this limit is not 0, the series diverges. 2. Limit comparison test Check to see whether La, is similar in appearance to a series Eb, whose convergence properties are known. and apply the limit comparison test. If lim a, /b, = L where L is finite and positive, then Ea, and Eb, either both converge or both diverge. 3. Ratio test If a, involves k!, ke, or at, try the ratio test, lim 24+1 = L. Converges if L 1, fails if L = 1. 4. Root test If it is easy to find lim Va, = L. try the root test. Converges if L 1, fails if [ = 1. 5. Integral test If f is continuous, positive, and decreasing, and a, = f(k) for all k, then ) a, and the improper integral [" (x) dx both converge or diverge. Think of using this test if f is easy to integrate or if a, involves a logarithm, a trigonometric function, or an inverse trigonometric function. 6. Direct comparison test If0 = a, s c, and 2 c, converges, then _ a, converges. If 0 s d, s a, and 2 d, diverges, then E a, diverges. 7. Zero-Infinity limit If lim b K = 0 and Eb, converges, then the series La, comparison test converges. a. Alternating Series them Alternating p-series converges for all positive p Consider Ilijan or ZEDkak , ak zo ( p > 0) K= 1 K = 1 If saks is decreasing and lim ak = o then alternating series converges