Limits of Sequences

Consider the sequence , which is . We see that the terms get closer to 1 as increases, and so we say that converges to 1 (and we may say that 1 is the limit of the sequence)

Notation

The limit as approaches is written:

(For infinity limits only)
Or

Important

Notice how the terms in the above sequence NEVER ACTUALLY REACH 1

When we say that , we are saying the difference between and becomes infinitesimally small as increases. Since can never actually equal , won't actually be equal to

Definition

A sequence converges to the limit for all positive numbers , there exists an integer such that:

Explanation: If the limit exists then we should be able to make as close as we want to by making large enough.
Conversely, if we can do that (for any distance , no matter how small), then the limit must exist.

Example

If , take . For any :

Observation: no matter how small is , we can always find an .

A well-writen proof for this looks like:

Example

Prove that as

For any , take

, so

Sequences Without Limits

If a sequence has no limit, it diverges. This can happen in a few different ways

The terms may grow without bound: e.g the limit of the sequence continues to grow larger and larger. The limit would be written .
There may be no pattern at all: e.g the digits of pi: . We still say the sequence diverges because it has no limit.
Some sequences oscillate between two numbers: e.g . We still say the sequence diverges.
It is possible for a sequence to contain subsequences which converge to different limits. e.g , the limits approach different values

left = -2; right = 10;
bottom = -10; top = 10;
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y = \sin(2\pi * n / 3) | DASHED
y = \frac{1}{10n + 1} | DASHED
y = \sin(2\pi * n / 3) + \frac{1}{10n + 1}