Part 1: Evaluating a series
Consider the sequence .
The limit of this sequence is
.
The sum of all terms in this sequence is defined as the the limit of the partial sums, which means
.
Enter infinity or -infinity if the limit diverges to or ; otherwise, enter DNE if the limit does not exist.
Part 2: Evaluating another series
Consider the sequence .
The limit of this sequence is
.
The sum of all terms in this sequence is defined as the the limit of the partial sums, which means
.
Enter infinity or -infinity if the limit diverges to or ; otherwise, enter DNE if the limit does not exist.
Part 3: Developing conceptual understanding
Suppose is a sequence.
If , then the series
choose
must
may or may not
cannot
converge. Hint: look back at parts 1 and 2.
If , then the series
choose
must
may or may not
cannot
converge.
If the series converges, then
choose
must
may or may not
cannot
be equal to .
You can earn partial credit on this problem.