Assignment:
Q1.) Find the interval of convergence of the series Σ (for n=0 to ∞) (4x-3)^(3n)/8^n and, within this interval, the sum of the series as a function of x.
Q2.) Determine all values for which the series Σ (for n=1 to ∞) (2^n(sin^n(x))/n^2 converges.
Q3.) Find the interval of convergence of the series Σ (for n=1 to ∞) (3^n (x-2)^n)/((the square root of (n+2)) 2^n)
Q4.) Suppose the interval of convergence of the Maclaurin series for f(x) is -2 < x < 2. If the Maclaurin series for (the integral from 0 to x) f(t) dt is obtained by integrating term-by-term, which of the following could be the interval of convergence of new series?
I. -2 < x < 2 III. -2 ≤ x < 2
II. -2 < x ≤ 2 IV. -2 ≤ x ≤ 2
a.) I only c.) II and III e.) I, II, III, IV
b.) IV only d.) I, II, III
Provide complete and step by step solution for the question and show calculations and use formulas.