Convergent Sequence

Algebra Level 3

$\large 1 + \dfrac 12 + \dfrac14 + \dfrac18 + \dfrac1{16} + \cdots + \dfrac1{2^n} + \cdots = \, ?$

The answer is 2.

1 solution

Kexin Zheng
Jun 27, 2016

This is a geometric sequence with $a_1=1$ and common ratio $r=\frac{1}{2}$ The sum of an infinite geometric series is $\frac{a_1}{1-r}$ Therefore, the sum is $\frac{1}{1-\frac{1}{2}}$ $=\boxed{2}$

You could also do it like this:

This series can be expressed as

$\frac { 1 }{ { 2 }^{ x } }$

Now, the first and the second term add up to 3/2.

The first and the second and the third add up to 7/4.

The first and the second and the third and the fourth add up to 31/16.

You see that it gets closer and closer to two, right?

$\therefore \lim _{ x\rightarrow \infty }{ \frac { 1 }{ { 2 }^{ x } } =2 }$

This means that the sum is 2.

alex wang - 4 years, 11 months ago

Convergent Sequence

The answer is 2.

1 solution

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