Let $\{X_n\}$ be a sequence and suppose that the sequence $\{X_{n+1} – X_n\}$ converges to $0$. Give an example to show that the sequence $\{X_n\}$ may not converge. Hence, the condition that $|X_n-X_m| < \epsilon$ for all $m,n \ge N$ is crucial in the definition of a Cauchy sequence.
Asked
Active
Viewed 398 times
2
-
Think of $x_{n+1} - x_{n}$ as the derivative of $x_n$. Does a function with a derivative approaching zero need to be bounded above? – Average Oct 14 '13 at 02:41
-
See also: Pseudo-Cauchy sequence, I want an example of a sequence that satisfies $|x(n) - x(n-1)| \to 0$ but not Cauchy, If ${x_n}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is ${x_n}$ a Cauchy sequence?, etc. – Martin Sleziak Dec 03 '16 at 07:31
2 Answers
3
Hint: What can you say about the partial sums of, say, the harmonic series?
Or a sequence involving $\ln{n}$?
More generally, think of your favorite function $f$ with $\lim_{n \to \infty} f(n) = \infty$, but that approaches $\infty$ "slowly."
1
Take the sequence $X_n = \sum_1^n \frac{1}{n}$. Here $X_{n=1} - X_n = \frac{1}{n}$ which goes to $0$ but $X_n$ is not convergent.
Supriyo
- 6,119
- 7
- 32
- 60