\[ 1+1-1+1 … = \frac{1}{2} \]
When one takes one from one plus one from one plus one and on and on but ends anon then starts again, then some sums sum to one, to zero other ones. One wonders who’d have won had stopping not been done; had he summed every bit until the infinite.
Lest you should think that such less well-known sums are much ado about nonsense I do give these two cents: The universe has got an answer which is not what most would first surmise, it is a compromise, and though it seems a laugh the universe gives “half”.
The above quote beuatifully describes the equation.
If we consider the above to be true on face value, we will be mugging up a formula, but not understand the inuitiveness of why infinite summations push us to think about deeper concepts in Mathematics. Another such case would be to go through the following example:
\[ 3/3 = 1 \]
Isnt it obviuos!
What about other fractions of \( 3 \)
\[ 1/3 = 0.333… \] \[ 2/3 = 0.666… \] \[ 3/3 = 0.999… \]
Wait!
Does this mean \( \frac{3}{3} \) is not equal to \( 1 \) but equal to \( 0.999… \)
Both are correct. It is accepted that \( 0.999… \) is equal to \( 1 \), which can be also phrased as \( 0.999… \) approaches \( 1 \) at infinity.
We can prove this with some simple steps.
\[ x = 0.999… \] \[ 10x = 9.999… \] \[ 10x = 9 + 0.999… \] \[ 10x = 9 + x \] \[ 9x = 9 \] \[ x = 1 \]
[Under Development…]