Chapter 9.4, Problem 42ES

(a) Show that the hypotheses of the integral test are satisfied by the series ${\displaystyle {\sum }_{k=1}^{\infty }1/\left(k^3+1\right).}$

(b) Use a CAS and the integral test to confirm that the series converges.

(c) Construct a table of partial sums for $n=10,20,30,\dots ,100,$ showing at least six decimal places.

(d) Based on your table, make a conjecture about the sum of the series to three decimal-place accuracy.

(e) Use part (b) of Exercise 36 to check your conjecture.