By working out these problems we can easily understand the topic.

Course Material 2.6 Some Important Limits

We may discuss some standard results stating the convergence of some sequences
which will be helpful in perusing Real Analysis.

1. If 0 ≤ 𝑟 < 1 lim 𝑟 𝑛 = 0
𝑛→∞

2. If −1 < 𝑟 < 1 then lim 𝑟 𝑛 = 0
𝑛→∞
1 𝑛
3. lim (1 + 𝑛) = 𝑒
𝑛→∞

4. If −1 < 𝑟 < 1 then lim 𝑛𝑟 𝑛 = 0
𝑛→∞
1
5. If 𝑎 > 0 then lim 𝑎𝑛 = 1
𝑛→∞
1
6. lim 𝑛𝑛 = 1
𝑛→∞
𝑎𝑛
7. lim = 0 for any 𝑎 ∈ ℝ
𝑛→∞ 𝑛!


Proof.

2. Since −1 < 𝑟 < 1 then we have 0 ≤ |𝑟| < 1 then by (1) lim |𝑟|𝑛 = 0
𝑛→∞

Therefore lim |𝑟 𝑛 | = 0
𝑛→∞

Hence given 𝜖 > 0 there exists a positive integer 𝑁 such that for all 𝑛 ≥ 𝑁
|𝑟 𝑛 | < 𝜖 this means that |𝑟 𝑛 − 0| < 𝜖, proving that lim 𝑟 𝑛 = 0
𝑛→∞

3. This limit defines the Mathematical constant 𝑒. We can prove that the sequence
1 𝑛
{𝑥𝑛 }, where 𝑥𝑛 = (1 + ) converges to a limit that lies between 2 and 3
𝑛

By the Binomial expansion, we have
1 𝑛 𝑛! 1
𝑥𝑛 = (1 + ) = 1 + ∑𝑛𝑘=1
𝑛 𝑘!(𝑛−𝑘)! 𝑛𝑘
𝑛
1 1 2 𝑘−1
=1+∑ (1 − ) (1 − ) … . (1 − )
𝑘! 𝑛 𝑛 𝑛
𝑘=1
We claim that {𝑥𝑛 } is increasing and bounded between 2 and 3.
1 1 2 𝑘−1
a. 𝑥𝑛 = 1 + ∑𝑛𝑘=1 𝑘! (1 − 𝑛) (1 − 𝑛) … . (1 − )
𝑛
𝑛
1 1 2 𝑘−1
<1+∑ (1 − ) (1 − ) … . (1 − )
𝑘! 𝑛+1 𝑛+1 𝑛+1
𝑘=1

< 𝑥𝑛+1
Hence {𝑥𝑛 } is increasing