@themathyam

1 year ago

CORRECTION: The formula for a geometric series I gave in this video is actually the formula for the series 1+x+x^2+....=1/(1-x). The formula I should have used is x+x^2+...=x/(1-x), so that the final calculation is

9(1/10+1/100+...)=9*(1/10/(1-1/10))=1

Sorry for that!

Original comment:

First of all, when we say "infinite sum", what we mean is that we take the limit of the sequence of partial sums

a_1,a_1+a_2,a_1+a_2+a_3,.... (We explain what we mean by limit below, the important thing is that this is a number).

 

Second, note that this shows that decimal expansions are NOT ALWAYS UNIQUE, which might be weird to those less familiar with math.

A common misconception for a lot of people who are exposed to the concept of limits for the first time is that a limit is some neverending proccess, and that we never 'reach' the actual limit. This is actually false, and the limit is defined exactly as the result of such a proccess, when such a result can be defined. Note that saying that a limit converges to a number is therefore inaccurare. A limit IS a number (when the sequence converges).

 

To be more accurate, we say a sequence of numbers a_1,a_2,... Converges to a number r if the sequence gets arbitrarily close to r if we go far enough (rigorously this is the epsilon-N definition of the limit). So our sequence approaches a definite, finite number, and it gets as close as we want. It then only makes sense to define the limit to be this number that our sequence gets arbitrarily close to.

 

Tying back to 0.99.., while the partial sums 0.99,0.999,0.9999 never reach 1, they get arbitrarily close to 1 if we add enough 9s. It then only makes sense that 0.999... is EXACTLY 1!

 

Some people argue that they are not the same because they should differ by 0.0000....1, where this 1 appears after "infinitely many zeroes". If we try to approach this with the definition of decimal expansion though, we see that all partial sums are zero (because there are, supposedly, "infinitely many zeroes" before the 1)

 

so that in fact

 

0.000...1=0

 

(In this case, the sequence is constant, so there is no confusion about what its limit is). Another way to see this is to consider the limit of

 

1-0.9, 1-0.99, 1-0.999 etc., and to check that it converges to 0.

 

Still confused? Feel free to reply below and I will answer to the best of my ability.❤

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