For what p does this converge?

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In summary, the limit [(lnB)^(-p+1)] / (-p+1) - [(ln1)^(-p+1)] / (-p+1) does not converge for any value of p. This is because ln(1) = 0 and when p is negative, the denominator would be 0, resulting in an infinite value. The statement in the book that states it converges for p>1 is incorrect.
  • #1
fk378
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Homework Statement


Given the limit (B-->inf) [(lnB)^(-p+1)] / (-p+1) - [(ln1)^(-p+1)] / (-p+1)

For what p does this converge?

The Attempt at a Solution



For the left side of the minus sign,
if 1-p<0 --> 0
if 1-p>0 --> inf

but for the ln(1) on the right side of the minus sign, ln(1)=0, so it would be
if 1-p<0 then you would get a zero in the denominator (since ln(1) would be raised to a negative power) and therefore ---> inf
if 1-p>0 then you would have zero as the numerator, but be left with infinity as the other value and hence ---> inf

The book says that for p>1 then it will converge. But how can that be, if you'll end up with a negative exponent for the 0??
 
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  • #2
Is that really "ln(1)" in the last term? ln(1)= 0 of course. Also parentheses would help.
Finally, you should not use "B" and "b" to mean the same thing but since there is no "b" in function I am going to assume you are. I am also going to assume that is [ln(b^(-p+1))]/[-p+1] - [(ln1)^(-p+1)] / [-p+1] which, since ln(1)= 0 is the same as ln(B-p+1/(-p+1)= (-p+1)ln(B)/(-p+1)= ln(b). So you have [itex]\lim{b\rightarrow \infty} ln(b) which does not converge for any p.

If you meant something else please clarify.

That converges to ln(B
 
  • #3
HallsofIvy said:
Is that really "ln(1)" in the last term? ln(1)= 0 of course. Also parentheses would help.
Finally, you should not use "B" and "b" to mean the same thing but since there is no "b" in function I am going to assume you are. I am also going to assume that is [ln(b^(-p+1))]/[-p+1] - [(ln1)^(-p+1)] / [-p+1] which, since ln(1)= 0 is the same as ln(B-p+1/(-p+1)= (-p+1)ln(B)/(-p+1)= ln(b). So you have [itex]\lim{b\rightarrow \infty} ln(b) which does not converge for any p.

If you meant something else please clarify.

That converges to ln(B

Please do not assume that when I stated (lnB)^(-p+1) I meant ln(B^-p+1) because I did not. The exponent is for the entire natural log expression. Therefore your statement regarding the answer to be just lnB is invalid. Thank you though.
 

Related to For what p does this converge?

1. What does "convergence" mean in this context?

In mathematics, convergence refers to the tendency of a sequence or a series of numbers to approach a certain value or limit as more terms are added.

2. How do you determine if a sequence or series converges?

The convergence of a sequence or series can be determined by evaluating its limit. If the limit exists and is a finite number, then the sequence or series is said to converge. Otherwise, it diverges.

3. What is the significance of "p" in this question?

The variable "p" represents the index or power of a sequence or series. It is used to determine if the sequence or series converges or diverges.

4. How can I find the value of "p" for which this sequence or series converges?

The value of "p" can be found by using various convergence tests, such as the ratio test, root test, or integral test. These tests can help determine the range of values for which the sequence or series converges.

5. Can a sequence or series converge for multiple values of "p"?

Yes, it is possible for a sequence or series to converge for multiple values of "p". This can occur when the series has different convergence behaviors depending on the value of "p".

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