Practice Infinity Intuition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

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

Numbers never stop—there's always a bigger one. Infinity isn't a number, it's a direction.

Example 1

medium
Show that the set of even positive integers can be put in one-to-one correspondence with the set of all positive integers, even though the evens seem 'smaller'.

Example 2

hard
Evaluate \displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n} and explain why an infinite sum can have a finite value.

Example 3

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?

Example 4

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