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Problem 23 Medium Difficulty

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} n!(2x - 1)^n $

Answer

$$I=\left\{\frac{1}{2}\right\}$$

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Video Transcript

figure out the radius of convergence. We figure out where we get convergence. So to figure out where we get convergence, we could do the ratio test on this thing by A N. I mean, this whole chunk here, including the X values So two X minus one to the n plus one. If we divide that by two X minus one to the end, then we're just going to be left with two X minus one. And then this will be the other stuff that we have after we do the algebra. Okay, And remember, in factorial is one times two times three times, not that all the way up to the Times, and so in, plus one factorial is going to cancel out within factorial. And the only thing that's going to be left over is in plus one. Okay. And for convergence, we're doing the ratio test here. We want for this to be something less than one. And now, if we we could notice that whatever we happen to put in for X here right as long as two x minus one is something that's finite and non zero two x minus one is non zero and we multiply it by something that blows up to infinity. Then we're going to get something that's infinite, an absolute value. So the only possible way to make this equality Hold this. This inequality holders, If two X minus one is equal to zero, we have to have that to X minus. One is equal to zero. And that's the only way that we're going to get this inequality tto be accomplished. Okay, so that means that two X is equal to one, so access one half. Okay, so the interval of convergence is just just the point, including one half just the interval, including one half here. Okay, so the length of this interval is one half minus one half. So this is this interval has no length to it. So that radius of convergence is zero and the interval of convergence is just the interval, including only one half

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