Let me get this straight: You want to know whether a series converges/diverges as it approaches the value x=2?
Assuming that is what you want to know, I give you this answer: Nearly all series will converge when it approaches 2. Why? You have to recall that series generally start at 0 or 1, and they increase by integers. This does not give time for a series to be divergent (meaning the sum approaches infinity) unless the series itself contains infinity. The reason you use improper integrals to determine convergence/divergence is because you can write a power series as a function of x.
If you have an asymptote in the middle of the function that represents a power series, then what you have to do is separate the integral into two integrals. Then, you evaluate the first with the upper limit approaching the asymptotic value from the negative side, and you evaluate the second with the lower limit approaching the asymptotic value from the positive side. Once done, your answer should condense into infinity or a finite number.
I suggest that you be careful with the accuracy of my second paragraph. The conditions for the Integral Test are that f is a positive, continuous, and decreasing function (maybe has to be differentiable over the interval too). For this reason, you would not be able to apply it to a function that has an asymptote. But by what I know of math, if you split the interval into two intervals where f satisfies the above condition on both intervals, then you'd be able to apply the Integral Test, but this time, in order for f to diverge, you'd only need one of the two integrals to diverge. Does that make sense? I kind of rumbled because series is not my forte.
