(a) Find the sum to infinity of the series 1-3x+5x² - 7x³ + .... ∞ where 0 < x < 1

1) Write an equation for determining the nth term of the series 20 + 10 + 5 + 2.5 + ... A) None of these are correct b) t(n) = 1/2n +20 c) t(n) = 1/2 (n−1)+20 d) t(n)=20 . (1/2)^n e) t(n) = 20 ⋅ 1/2^n-1

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(3n)! Use any method to determine if the series (n!)3 n=1 converges or diverges. Show your work.

4. (6 points) Explain how to reorder the terms of series sums to 2022. n=1 (-1)" n so that the resulting

5. Consider the series n! n=0 for real numbers x. (a) Use the ratio test to determine the values x for which the series is absolutely convergent. (b) Does this mean that the series is convergent for these values of x? Explain. (c) Explain how one can use part (b) above to determine the limit of the sequence {an} where 22n an n!