Sequences — Question 3

*Question 3 Let $a_n=\dfrac{n}{n+1}$ for $n\ge1$.

1. Prove algebraically (without derivatives) that $(a_n)$ is strictly increasing.

2. Find an upper bound and use the Monotone Convergence Theorem to explain why a limit exists.

3. Find the limit and express the error $|a_n-L|$ exactly.

*Question 3 – Solution

Compare consecutive terms by subtraction: $$a_{n+1}-a_n=\frac{n+1}{n+2}-\frac n{n+1}=\frac1{(n+1)(n+2)}>0.$$ The denominator is positive for $n\ge1$, so $a_{n+1}-a_n>0$ and $(a_n)$ is strictly increasing.

Rewrite the term as $$a_n=1-\frac1{n+1}<1,$$ Thus $a_n<1$ for every $n$, so the sequence is bounded above by $1$.

An increasing sequence bounded above converges by the Monotone Convergence Theorem. Using the rewritten formula, $$\lim_{n\to\infty}a_n=\lim_{n\to\infty}\left(1-\frac1{n+1}\right)=1.$$

Since $a_n<1$, the exact error is $|a_n-1|=1-a_n=1/(n+1)$, which tends to zero.