Chapter 7: Q. 1 (page 614)
Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
Part (a): If the sequenceconverges, then the seriesconverges.
Part (b): If a series diverges and is its sequence of partial sums, then .
Part (c): If two series localid="1649656219815" both diverge, then the series diverges.
Part (d): If are two convergent series, then for any real numbers c and d, .
Part (e): If the series converges to , then the series converges to a value .
Part (f): If the series converges, then .
Part (g): If a geometric series converges, then .
Part (h): The series where for , converges to.
Short Answer
Part (a): The given statement is false.
Part (b): The given statement is false.
Part (c): The given statement is false.
Part (d): The given statement is true.
Part (e): The given statement is false.
Part (f): The given statement is true.
Part (g): The given statement is true.
Part (h): The given statement is false.