Why does the ln(x) series converge for x=2 but not x=0?

In summary, the Taylor Series for ln(x) centered at 1 converges for the endpoint x=2 but not x=0 of the interval of convergence. This is because when x=2, the resulting alternating series converges by the alternating series test, while when x=0, the resulting series does not alternate and therefore does not converge. This mistake was due to forgetting to multiply n by 2 when determining the sign of the terms.
  • #1
schaefera
208
0

Homework Statement


This isn't really so much a homework problem as me asking a question. The Taylor Series for ln(x) centered at 1 is: sum_[0, infinity] ((-1)^((n+1)*(x-1)^n))/n, then why does it converge for the endpoint x=2, but not x=0 of the interval of convergence?

Homework Equations


The Attempt at a Solution


Letting x=2, you get the alternating series (-1)^(n+1)/(n), which converges by the alternating series test.
Letting x=0, don't you get (-1)^(2n+1)/n, which should also converge as an alternating series?
 
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  • #2
(-1)^(2n+1) does not alternate. It always has the same sign.
 
  • #3
And now I just feel silly! I'm so used to a series having a (..+1) in the exponent to alternate, looks like I forgot how to multiply n by 2! This is what happens when I have to worry about infinity.

Thanks!
 

FAQ: Why does the ln(x) series converge for x=2 but not x=0?

What does "convergence of ln(x) series" mean?

The convergence of ln(x) series refers to whether or not the infinite series of ln(x) converges to a finite value or diverges to infinity. In other words, it determines if the sum of all the terms in the series has a definite limit or not.

How do you determine the convergence of ln(x) series?

To determine the convergence of ln(x) series, you can use various tests such as the ratio test, the root test, or the integral test. These tests analyze the behavior of the terms in the series to determine if they approach a finite value or not.

What is the significance of the convergence of ln(x) series?

The convergence of ln(x) series is important in mathematics as it allows us to evaluate the sum of an infinite series and determine the behavior of the terms within the series. It also has applications in various fields such as physics, engineering, and finance.

Can the convergence of ln(x) series change?

Yes, the convergence of ln(x) series can change depending on the value of x. For example, the series may converge for some values of x but diverge for others. It is important to note the range of values for which the series converges.

What is the difference between absolute and conditional convergence of ln(x) series?

Absolute convergence of ln(x) series means that the series converges regardless of the order in which the terms are added. On the other hand, conditional convergence means that the series only converges when the terms are added in a specific order. In other words, the rearrangement of terms can affect the convergence of a conditionally convergent series.

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