Hello! The following question has been tripping me up a little bit:
"Show that the sequence \(\displaystyle a_n=\dfrac{n}{n^2+1}\) eventually stays within a distance \(\displaystyle \dfrac{1}{10,000}\) from 0." Here is my working:
If the sequence stays within a distance \(\displaystyle \frac{1}{10,000}\) from 0, there exists N such that \(\displaystyle \forall n>N\),
\(\displaystyle |\frac{n}{n^2+1} - 0|<\frac{1}{10,000}\)
I then solve this inequality to get that if we take N=10,000, the sequence will stay within the required distance. But I don't really feel like I've shown this, and even if I have, it's a very inelegant and clunky method that I feel like there's probably a quicker way which is eluding me, and I feel sad for being unable to see this.
Any help would be appreciated.
"Show that the sequence \(\displaystyle a_n=\dfrac{n}{n^2+1}\) eventually stays within a distance \(\displaystyle \dfrac{1}{10,000}\) from 0." Here is my working:
If the sequence stays within a distance \(\displaystyle \frac{1}{10,000}\) from 0, there exists N such that \(\displaystyle \forall n>N\),
\(\displaystyle |\frac{n}{n^2+1} - 0|<\frac{1}{10,000}\)
I then solve this inequality to get that if we take N=10,000, the sequence will stay within the required distance. But I don't really feel like I've shown this, and even if I have, it's a very inelegant and clunky method that I feel like there's probably a quicker way which is eluding me, and I feel sad for being unable to see this.
Any help would be appreciated.