- #1

DMOC

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## Homework Statement

Determine values for x where [tex]\sum \frac{(2^k)(x^k)}{ln(k + 2)}[/tex] converges if k goes from 0 to infinity.

## Homework Equations

Limit test and alternating series test.

## The Attempt at a Solution

I used the limit comparison test to test out a

_{k+1}and a

_{n}and I ended up getting [tex](-1/2) \le x \le (1/2)[/tex]

Then I had to compare the endpoints. I noticed that the numerator is always 1 in the case of x=1/2 or always 1 or -1 (in the case of x = -1/2).

So by the alternating series test, I concluded that BOTH of the endpoints converges. However, my answer key says that x=1/2 diverges while x=-1/2 converges (I'm correct there).

FYI, for x = 1/2,

I said that a

_{n+1}< a

_{n}and that the limit as k -> [tex]\infty[/tex] = 0, so why does it apparently diverge?