Determine if the following sequence converges. If it does, find its limit. 8 √2n! \ (2n)! (n=1 2. Determine if the following sequence converges. If it does, find its limit. 210nn! (2n)! n=1 8 3. Let x be a (constant) real number. Determine, with proper justification, if the sequenc- 8 27°⁰ cos(nx) n³ converges. If it does, find the value of the limit. n=1

Find the limit of the sequence. Hint: Remember that cos n always satisfies -1 ≤ cos n ≤ 1. an = (cos n) 7n

Use a graphing utility to graph the first 10 terms of the sequence with the given nth term. nπ 3 = COS an Use the graph to make an inference about the convergence or divergence of the sequence. The sequence converges. The sequence diverges. Verify your inference analytically and, if the sequence converges, find its limit. (If the quantity diverges, enter DIVERGES.)

1. Determine if each sequence converges or diverges. Find the limit of any convergent sequences. 4+n-3n² 4n²+1 (a) { (b) {ln(12n+2) - In(9+4n)} (2n-1)! n! (c) {. (d) cos(nπ) 2} n (e) {(1 + ²)"} (f) {n(√n² +1 − n)} 82n n!

Use the method in Example 3.3.5 to calculate Sqrt(2), correct to within 4 decimals.

Let a, = 2" cos(n7) for n21. Write out the first 8 terms of the sequence (a, through ag). Determine lim a, . Note: You must show all your work and justify your steps carefully. а. b. Determine if Ea, converges (absolutely or conditionally) or diverges. Note: You must show с. all your work and justify your steps carefully.