So someone told me its infinity, but i don't get it
it sure looks like its just keep getting smaller and smaller so how is this going to infinity? or is it?
I know that 1+1/2+1/4+1/8+... is 1 so how is this infinity?
Asked
Active
Viewed 131 times
0
-
1The harmonic series is divergent: $a_n\to 0$ is not enough to ensure $\sum a_n<\infty$. – Jack D'Aurizio Jan 15 '18 at 14:58
-
1The sum $1+1/2+1/4+1/8+\ldots$ is $2$, not $1$. – Ross Millikan Jan 15 '18 at 15:04
2 Answers
4
$$ \dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6}+\dfrac{1}{8}+\cdots=\dfrac{1}{2}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\cdots\right) $$
VJunior
- 807
-
Wow that's actually a very simple and great answer, i googled and the previous answers on stackexchange were so vague. Thank you. also for those who might not know, the sum of 1+1/2+1/3+... is infinite so this is why. – Richard Jones Jan 15 '18 at 15:03
1
No, sorry. Note that the sum: $$S = 1 +\frac12 +\frac14 + \frac18 +\ldots$$ converges to $\color{red} 2$. A typo perhaps?
Now, the sum: $$S = \frac12 + \frac14+\frac16+\ldots$$ $$= \frac12 \left[1 + \frac12 + \frac13 + \ldots \right] $$ $$= \frac12 S_1$$ where $S_1$ diverges as was shown here on MSE earlier.