MHB Why Is the Limit of This $\ln$ Sequence Incorrect?

Click For Summary
The sequence in question is defined as \( a_n = \ln\left({\frac{n}{n^2 + 1}}\right) \). The limit being evaluated is \( \lim_{{n}\to{\infty}} \ln\left({\frac{n}{n^2 + 1}}\right) \), which involves first finding the limit of \( \frac{n}{n^2 + 1} \) as \( n \) approaches infinity. While this limit approaches 0, substituting this directly into the logarithm leads to \( \ln(0) \), which is undefined and results in negative infinity. The error lies in the incorrect assumption that the limit of the logarithm can be determined simply by evaluating the limit of its argument without considering the behavior of the logarithm near 0. Therefore, a more careful analysis of the logarithmic function's limit is required.
tmt1
Messages
230
Reaction score
0
If I have this sequence

$$a_n = \ln\left({\frac{n}{n^2 + 1}}\right)$$

I need to find:

$$ \lim_{{n}\to{\infty}} \ln\left({\frac{n}{n^2 + 1}}\right)$$

Shouldn't I be able to find the limit of$$ \lim_{{n}\to{\infty}} \frac{n}{n^2 + 1}$$ (which is $0$) and then substitute the result of that into the original limit and get the answer there?

So if I substitute in 0 I would get

$$ \lim_{{n}\to{\infty}} \ln\left({0}\right)$$

which would be negative $\infty$. However this is the incorrect answer.
 
Last edited:
Physics news on Phys.org
Why do you think that the result is incorrect?
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
View full post »

Similar threads

Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad A correct definition of sequential right continuity of a function
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Simplifying limit with Stirling approximation
  • · Replies 9 ·
Replies
9
Views
2K
MHB Evaluating Limit $$\frac{\ln2}{2}+\cdots+\frac{\ln n}{n}$$
  • · Replies 2 ·
Replies
2
Views
1K
MHB Improper Integral: Definition and Evaluation
  • · Replies 8 ·
Replies
8
Views
1K
MHB Is the Convergence of These Sequences Correctly Determined?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
MHB Evaluate Limit: $\frac{0}{0}$ - Can Someone Help?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad What is the limit of (a^n)/n for a>1?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
2K