Chapter 10.5, Problem 63E

By repeatedly differentiating the exponential series e* = #=0 find the sum S="² n= n! n! and then setting x = 1,

Suppose a power series converges if | 4x - 8 | ≤ 40 anddiverges if | 4x - 8 | > 40. Determine the radius and interval of convergence.

What is the limit as n goes to infinity of the power series (log n)2/nxn ?

Find the Taylor series about the point x0=2 for the function 1/(1-x).

How do I solve for this? Is there a way to show the limit of the sequence and its convergence without using other series tests?

Find the value of the integral 8x(In(7x))² ' Determine whether the series 1 8n(In(7n))? n=2 is convergent. Enter C if the series is convergent, D if the series is divergent.

Find a power series for the function f (x) = x2/(1 + x3)

Please use these questions to find non-zero terms!

For the included problem, I keep getting that the series converges, but the correct answer is that it diverges. I used both the limit comparison test and the root test and got that it converges both times with these tests. How does the series diverge and what test should I use? Σ(k=1 to infinity) of (2k+k)/(4k^4-8k)