Question

On finding the interval of convergence for a power series, I need help testing an endpoint. Endpoint x= -1 for ∑(1/n2^n)*(x-1)^n (which would be -2^n/n2^n) ....what test would I use to test the endpoint? Please show work.

Answer 1

\[\sum{ \frac{ (-2)^n }{ n2^n }} = \sum{ \frac{ (-1)^n2^n }{ n2^n }}=\sum{ \frac{ (-1)^n }{ n }}\]

and then use the alternating series test from here, does that help?

Answer 2

IT would be 1/n? so it diverges???

Answer 3

Note the \[(-1)^n\] which makes it alternating.

Answer 4

so the lim as n goes to infinity of (-1)^n/n would be 0....so it converges?

Answer 5

yes, it converges but you must show that with the alternating series test. Do you know that test?

Answer 6

yes, it converges if it is a decreasing sequence and the lim equals zero