Ah! okay, need solving 14, and just checking for 16 and 4.

4. i got the series is divergent by root test because you get e > 1

14. i was thinking maclaurin series? but it isn't working out properly?

16. i need to find a power series representation and interval of convergence for the given series. i found the power series representation to be:
(-1)^n(n+1)x^n and the lim by ratio test comes to abs value of x. should i be testing endpoints?

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.