Vn! 1. Compute lim and E 2. Evaluate the sums without using power series. n1...

Transcribed Image Text:

Vn! 1. Compute lim ž and E 2. Evaluate the sums without using power series. n1 2" n=1 2" (2n)! =o 4°(n)?(2n + 1) 3. Does the sum converge or diverge: ? (Another way of asking this problem is: find the interval of convergence for the power series for arcsin(x) centered at 0.) 4. Show that given any conditionally convergent series, it is always possible to rearrange the order of the terms to make the series add up to 42, but that for any absolutely convergent series, rearranging never changes the sum. 5. Let fx) be the function obtained from e7 by filling in the hole at the origin (that is, declare f(0)-0). Compute the Maclaurin series for f(x). Show that it converges, but not to fix). [The difficulty in this problem is that here is no a priori guarantee that (0) = limf(x), so you will need to use the definition of derivative as the limit of the difference quotient.) Vn! 1. Compute lim ž and E 2. Evaluate the sums without using power series. n1 2" n=1 2" (2n)! =o 4°(n)?(2n + 1) 3. Does the sum converge or diverge: ? (Another way of asking this problem is: find the interval of convergence for the power series for arcsin(x) centered at 0.) 4. Show that given any conditionally convergent series, it is always possible to rearrange the order of the terms to make the series add up to 42, but that for any absolutely convergent series, rearranging never changes the sum. 5. Let fx) be the function obtained from e7 by filling in the hole at the origin (that is, declare f(0)-0). Compute the Maclaurin series for f(x). Show that it converges, but not to fix). [The difficulty in this problem is that here is no a priori guarantee that (0) = limf(x), so you will need to use the definition of derivative as the limit of the difference quotient.)