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These are practice problems for series and power series. For all of the following series check (and fully justify why) whether the series converges or diverges. For problems with a star, if the series converges find its sum. If the problem is about a power series find both its radius and interval of convergence 1. ∞ X 2 ln n n=1 2. ∞ X cos2 (5n) n=1 3. 4. 5. ∞ X 7 n5n n=1 ∞ X ∞ X 8. n3 n3 2n n! ∞ X 8n 32n n=1 ∞ X n=1 9. 4+ 1 √ 4 (−1)n n=1 7. n3 ∞ X (−4)n n n 6 n=1 n=1 6. n6 2n2 9n2 − 7 ∞ X 10 + 9n n=1 1 5 + 8n 10. 11. ∞ X 6n 6n+4 n=1 ∞ X 9n + 2n 12. ∞ X (−1)n n=1 13. [∗] 10n n=1 ∞ X n3 + 1 n3 − 7 n!(x − 4)n n=1 14. ∞ X (−1)n n7n xn n=1 15. 16. ∞ X xn (n + 9) (−1)n 5n n=1 ∞ X (−1)n n=1 cos n n5 Hint: it converges; a bit tricky, you need to apply two tests one after the other 2