pls explain?

Question

let d be the region in the xy plane in which the series {sigma (infinity,k=1) ((x+2y)^k)/k converges.then the interior of d is
1)an open disk
2)the open region bounded by an ellipse
3)the open bounded by a quadilateral
4)the open region between 2 parallel lines
5)an open half plane

Answer 1

By the ratio test, we know that it converges wherever:

lim (k-> inf) |x+2y| (k / (k+1) ) < 1

that means |x+2y| < 1

That is an open region between two parallel lines:

(-1-x)/2 ≤ y < (1-x)/2

The interior would be an open region, but it's interesting to note that the series diverges on the upper bound (it reduces to the harmonic series), but converges on the lower bound (reducing to the alternating harmonic series).