01
Power series - radius and interval
Find the radius and interval of convergence for these power series:
(a)
(b) (c)
Solution
01
(a)
Therefore, the radius of convergence is
and the preliminary interval is . Check end points:
Both of these series converge, so the final interval of convergence is
.
(b)
Therefore,
and the preliminary interval is . Check end points:
The first series converges by the AST. The second diverges (
). So the final interval of convergence is
.
(c)
Therefore, the radius of convergence is
and the preliminary interval is . Check end points:
Both series diverge. So the final interval is
Link to original.
02
Power series - radius and interval
Find the radius and interval of convergence for these power series:
(a)
(b) (c)
Solution
06
(a)
Therefore
and .
(b)
Therefore
and the preliminary interval is . At
we have . This diverges ( ). At
we have . This converges by the AST. Therefore, the final interval of convergence is
.
(c)
Observe that
so . Assume . Then: If
, then of course the series is and converges to . Therefore
Link to originaland .
03
Power series - radius and interval
Find the radius and interval of convergence for these power series:
(a)
(b)
Solution
07
(a)
Apply the ratio test:
Therefore
and the preliminary interval is . Check endpoints:
At
, we have , which converges absolutely. At
, we have , which converges by the DCT, comparing with . Therefore, the final interval of convergence is
.
(b)
Apply the ratio test:
Therefore,
Link to originaland the interval of convergence is .