Infinity Intuition Math Example 2

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

hard

Evaluate

\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n}

and explain why an infinite sum can have a finite value.

1
Write the partial sums: $S_N = \dfrac{1}{2} + \dfrac{1}{4} + \cdots + \dfrac{1}{2^N} = 1 - \dfrac{1}{2^N}$ (geometric series formula).
2
As $N \to \infty$ , $\dfrac{1}{2^N} \to 0$ , so $S_N \to 1$ .
3
The infinite series converges: $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{2^n} = 1$ .
4
Intuitively: halving, then halving again, infinitely many times fills exactly one whole — like halving a bar of chocolate forever.

Answer

\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{2^n} = 1

An infinite series can converge to a finite value when the terms shrink fast enough. The geometric series

\sum r^n

converges if and only if

|r| < 1

, giving sum

\frac{a}{1-r}

. This resolves Zeno's paradox: infinitely many steps can be completed in finite time if each step is small enough.

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

The sequence [formula] keeps halving. Does this sequence have a last term? What value does it approa

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