Check the convergence of the following series: k=1 ak²+bk+c dk² + ek + f where a, b, c, d, e, and f are fixed natural numbers. k

a) Determine whether each of the series below converges or diverges and explain your reasoning (k!) ³ (3k)!

Use Ratio Test to check the convergence of the series. Σ n2 1 1+ n n=1

Decide if each infinite geometric series is convergent or divergent. State the sum of the 14. series, if it exists: a. -64+16-4+1-... - - - -- 12 10 +.. b. 24 6.1 + 1.22 + 0.244 + 0.0488 +.. d. 12 +30 75+.. c.

19. Σ n!(2x − 1)" -1

Write the following series using sigma notation. 1 1 1 1 (e) -++-V- x4 1 1 (b) 1+ x2 2 (а) 1 — + 2 3 4 6 4 Hint: look for common patterns like alternating signs, powers, factorials, even or odd numbers, etc.

No AI Solution

3-22 Find the radius of convergence and interval of conver- gence of the series.

Define a digit to be an element of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Show that for any sequence {a} of digits, the series [ converges. n=1 10" n=1