Can someone please help
2 answers:
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4. Correct. You also could have used the limit test for divergence for the same conclusion (the summand approaches infinity).
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14. I'm guessing the instructions are the same as for 16. Rewrite as
Now recall that for
, we have
so that for this function, we get
Because this is a geometric sum, this converges when
, or 
. This would be the interval of convergence.
Your hunch about checking the endpoints is correct. Checking is easy in this case, because at the endpoints (-3/2 and 3/2) the series obviously diverges.
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16. This one is kind of tricky, and there's more than one way to do it. The standard method would be to take the antiderivative:
We also have
and differentiating this gives
By the ratio test, this converges when
The limit reduces to
and so the series converges absolutely for. Checking the endpoints is also easy in this case. The factor of 
is a clear sign that the series will diverge at either extreme.
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14. I'm guessing the instructions are the same as for 16. Rewrite as
Now recall that for
so that for this function, we get
Because this is a geometric sum, this converges when
Your hunch about checking the endpoints is correct. Checking is easy in this case, because at the endpoints (-3/2 and 3/2) the series obviously diverges.
- - -
16. This one is kind of tricky, and there's more than one way to do it. The standard method would be to take the antiderivative:
We also have
and differentiating this gives
By the ratio test, this converges when
The limit reduces to
and so the series converges absolutely for