Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges...

Question

Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.

e. Diverges for x = 8

Accepted Answer

Hello there, today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. True or false, if the power series, the sum of a n multiplied by x to the power of n, converges at x is equal to -4 and diverges at x is equal to 6, then it also diverges at x is equal to 7. OK. So it appears for this particular prom we're given a statement, and the statement once again is if the power series, the sum of a n multiplied by x to the power of n, converges at x is equal to -4 and diverges at x equal to 6, then it also diverges at x equal to 7, and we're asked to determine if this statement will be a true orb false. So now that we know what we're ultimately trying to solve for, we need to recall and note the following. So based on our prior knowledge, we need to recall and note that for a power series, the sum of a n multiplied by x to the power of n centered at 0, there will be a radius of convergence. R is greater than or equal to 0, and for such a series, it will converge for all x such that the absolute value of x is less than R. And it will diverge for all X such that the absolute value of X is greater than capital R. And the behavior at the absolute value of X is equal to R must be checked separately. So that means, based on our prior knowledge and what we're told by the problem itself, that convergence, so I'll say C for short, so convergence at X is equal to -4. So when we have convergence at x is equal to -4. It will give us the absolute value of 4 is equal to 4, which is greater than, so once again, we need to note that the absolute value of -4 is equal to 4, and 4 will be less than or equal to R, which I'll just call R for short. So once again, anything inside of absolute value signs becomes positive, so the absolute value of -4 is equal to 4 and 4 is less than or equal to R. Thus, we can go ahead and note that R is greater than or equal to 4. Thus, the smallest possible r, which we'll call our subscript min, so R min will be equal to simply 4. So R min is equal to 4. So when we have divergence, which I'll call D for short, so divergence at x is equal to 6. Will give us the absolute value of 6 is equal to 6 and 6 is greater than r, and we need to note, and let's make a note here off to the side that if r Is greater than 6, so once again. Here, let's make a note that if we have R is greater than 6, so let's put a giant if, so if, if, once again, R is greater than 6, the series would converge at, so it will converge, which I'll put a C in quotes, so it will converge at X is equal to 6. Thus, that will mean we can safely write that R is less than or equal to 6. So once again, R is less than or equal to 6. Thus, as a result of this fact, Our max, so once again, our subscript max will be equal to 6. So when we combine these bounds, it will give us that R is an element of the closed interval square brackets of 4.6, and this represents the maximum possible interval of convergence. And we need to note if the absolute value of X is less than 4, then that will mean as a result, the absolute value of X is less than R, and this will be for every, so this will be for. Every, and let's make a note in our note in our. Notes here that every R will be an element of the closed interval square brackets of 4.6, so that means the convergence at such an x value is guaranteed. And similarly, if the absolute value of x is greater than 6, then that will mean the absolute value of X is greater than R, and this will be for every R is an element of the closed interval square brackets of 4,6. And that will mean that the divergence at such an X value is guaranteed. And if 4, so once again, if 4 is less than or equal to the absolute value of X, which is less than or equal to 6, that means nothing definite can be concluded from the given information alone. So whether the series converges there depends on the actual R within our closed interval 4.6, and on the pointwise boundary behavior. So that means we need to apply this to X is equal to 7. So the absolute value of 7 is equal to 7 and 7 is greater than 6. And R is less than or equal to 6, thus, that means that 7 is greater than R. Thus, the series must diverge at, so once again it will diverge. At X is equal to 7, so therefore that will mean that our statement without a doubt is true. Hooray, we did it. So going back to look at our multiple choice answers, our correct and final answer based on the information we've discovered together and our prior knowledge, means that our final answer will be a true. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.e. Diverges for x = 8

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Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
e. Diverges for x = 8