Infinity Intuition Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

The sequence

1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}, \ldots

keeps halving. Does this sequence have a last term? What value does it approach?

1
There is no last term: given any term $\dfrac{1}{2^n}$ , the next term $\dfrac{1}{2^{n+1}}$ is always smaller but still positive.
2
As $n \to \infty$ , $\dfrac{1}{2^n} \to 0$ . The sequence approaches $0$ but never reaches it.

Answer

There is no last term; the sequence approaches

0

without ever equalling

0

An infinite sequence can get arbitrarily close to a limit without reaching it. This is the essence of convergence:

\frac{1}{2^n}

gets smaller than any positive number you name, yet it is always positive. The limit

0

is a boundary the sequence approaches, not a value it attains.

About Infinity Intuition

The concept of endlessness or unboundedness—a process that goes on forever with no final stopping point.

Show that the set of even positive integers can be put in one-to-one correspondence with the set of

Evaluate [formula] and explain why an infinite sum can have a finite value.

Compare: is there 'more' natural numbers or 'more' integers? Explain using a bijection.