Why do you think it's dubious the way they added the partial sums? All they did is multiply each term by something that is equal to 1 as N goes to infinity (notice that cos(n/N) is 1, since n is constant and N goes to infinity and cos(0)=1), so there's nothing weird or wrong about that. If you could elaborate on your doubt I may be able to explain it better

Thanks! I'm glad you liked it!

That's my secret cap, I'm always confused

Bonehead: "In base 2 ".11..." has a limit of 1 In base 3 ".22..." has a limit of 1 ETC..." Nonsense. Decimals don't have limits, they have sums. The sum of 0.111... (base 2) is 1. The sum of 0.222... (base 3) is 1, etc. Bonehead: "Writing out the series in base 10 Base 2 1/2, 1/4, 1/8, 1/16,... 1/2^n Base 3 1/3, 1/9, 1/27,... 1/3^n Base 10 1/10, 1/100,... 1/10^n Base 16 1/16, 1/256... 1/16^n Base 60 1/60, 1/360... 1/60^n ETC..." What was any of that about? You wrote some sequences, not series, and you did that bizarrely. Quelle surprise. The first sequence is conventionally written as 1/2, 1/4, 1/8, ..., 1/2^n, ... The other sequences are written in a similar way. Bonehead: "According to many maths the sum of each of those series is exactly 1." What series? If you mean e.g. 0.bbb... (base b+1) has the sum 1, then you'd be right. Bonehead: "I disagree, the sum of each of those series is the limit of the sum of their partial finite sums." You can't even remember the phrase (actually definition): the sum of a series is the limit of the sequence of its partial sums (if the limit exists). So in fact, you are agreeing with the mathematicians. It has been evident for quite a long time that you are such a humongous muppet that you cannot tell the difference between a series, such as 0.999..., and a sequence, such as 0.9, 0.99, 0.999, ... even though you have been at it for at least a year. You aren't even referring to the correct sequences. Sticking with 0.999... = 9/10 + 9/10^2 + 9/10^3 + ... + 9/10^n + ... you (badly) defined a sequence 9/10, 9/10^2, 9/10^3, ..., 9/10^n, ... and then reconstructed the series 0.999... as lim n->oo Sum(k = 1 to n, 9/10^k) = lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1. You are such a muppet that you haven't realised that you are agreeing with the mathematicians. No doubt you'll now do your usual trick and back pedal on what you said in your comment, perhaps after going silent for a few days. Bonehead: "Also the true value of each of those bases geometric series sum gets closer to 1 as the base goes higher." All those geometric series have the sum 1. They are constant series and so cannot get closer to anything, as that requires change. You continue to fundamentally mistake a series for a sequence. One year (or more) and you still haven't understood what most people understand in, at most, a few seconds.

"0.999..." is a representation of the same real number that "1" represents. As 0.999... = 1, of course there is only one real number in the equality.

@@Chris-5318 Yes of course but what do you think about people who believe 0.99.. is a real number "by itself" In french, the terminating decimal representation is said to be proper and the alternative representation (as a repeating decimal whose repetend is the digit 9) is said to be improper (another way to say uncorrect ?). The second one seems to be more a code than a number by itself. Don't you think so ?

@@cupiodissolvi9942 "by itself" 0.999... is just a shorthand for a formal power series: a priori, it does not have a value. In English a terminating decimal is said to be a finite (terminating) or infinite (non-terminating). Neither is correct nor incorrect. Infinite decimals (and series) are different in that notions such as convergence are used with them. In essence, the sum of infinitely many terms is just a natural extension to the sum of finitely many terms. I don't recall a time when I ever had a problem dealing with infinite series (in principle). The generalisation of sum is: the sum of a series is the limit of the sequence of its partial sums (if the limit exists). So: [the sum of] 0.999... := lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1 The last equality follows from the definition of limit.

Basics say that a real number is unique and precise and can be represented as all the points in a s straight line. On first impression, representation and equality are easily confused. Based on how we use decimal digits to represent real numbers some maths find it hard to deal with infinity and decimal digits to represent real numbers without further justification. ".33..." is incomplete division to represent the ratio of 1 to 3 or the fraction 1/3.

@@johnlabonte-ch5ul I do understand the 0.33... notation as a consistent notation. But 0.999... is not consistent as a decimal notation. It is only consistent as a sum and its limit. Don't you agree ?

(...continued) So how does this linear relationship in the form of y = mx relate to the rotations, the trigonometric functions, the i, or the complex values? Well, let's simplify this expression of y = mx even a little more by applying the multiplicative identity property of (a * 1 = a) for all (a) to the slope m. Here we can let m = 1 and through this property x is unchanged. This leaves us with the expression y = x. Here this expression alone which can be treated as both assignment and equality. For all values of X in X, Y is also equal to X. Here X remains unchanged. This is equivalent to adding 0 to any and every element in X to get Y through the use of the additive identity property. This expression when plotted by the equivalent pairs { ..., (-1,-1), (0,0), {1, 1}, ... } gives use the line that bisects the XY plane in both the 1st and 3rd quadrants. We know that this line has a 45 degree or PI/4 radian angle both above and below it gradient between X and Y. We know this and that it is self-evident for several reasons. First, we know that the X and Y axes are perpendicular, orthogonal to each other as they create a Right Angle at the Origin which is 90 degrees or PI/2 radians respectively and dividing them by 2 gives us 45 degrees and PI/4 radians. A slope of 1 is 45 degrees or PI/4 radians. We can also see this from evaluating or solving the slope formula for any two points. Here, I'll just use the origin (0,0) and (1,1) to illustrate this in the general case. (y2-y1)/(x2-x1) = deltY/deltaX = (1-0)/(1-0) = 1/1 = 1. From this we can also represent or substitute deltaY for sin(t) and deltaX for cos(t) where t or theta is the angle between the line of y = mx+b and the +x-axis. With these we can also define the slope formula for m as the following: m = sin(t)/cos(t) = tan(t). We use this form of the slope when the value of the angle is known. m = sin(45)/cos(45) = tan(45) = 1. So where does division of 0 come into play? We are about to see this in translational action by evaluating the slopes or gradients of linear expressions with respect to their angle of rotation. We have been taught that when a slope of a line is 0, that it is horizontal or that all horizontal lines have a slope of 0. This true. We can see this by the following: m = sin(0)/cos(0) = 0/1 = tan(0) = 0. We are also taught that division by 0 and tan(90) are undefined. This is one of the notions that I want to challenge as well as the common nomenclature of calling i and multiples of i, the imaginaries and we will see why shortly. But first, we need to understand the relationship between the angles and the slopes. This table should help. range of angles | range of slopes t = 0 | 0 0 < t < 45 | 0 < m < 1 t = 45 | m = 1 45 < t < 90 | m > 1 : limit extends to +infinity within the first quadrant. The mirror or reflection for negative slopes is also true within the 3rd quadrant approaching negative infinity. t = 90 | m = ? : This is where we are taught that it is undefined. This is what I'm going to challenge. We saw that when t = 0, the slope is also 0 through sin(0)/cos(0) = 0/1. We are seeing this as a fraction by division which is appropriate however, when t = 90 and we have sin(90)/cos(90) which gives us the reciprocal of 0/1 being 1/0, now all of a sudden there appears to be an issue and we can't do this? I beg to differ. If we take the fractional values of the slope a/b or deltaY/deltaX and treat them as such that they are the vectors or coordinate pairs (deltaX, deltaY) or (cos(t), sin(t)) everything will begin to make proper sense. Oh, wait a minute, I've seen this notation before. These are the coordinate pairs for the Unit Circle. So why is division by 0 and tan(90) not undefined? The issue here is that they are actually well defined, it's just that they are no longer a 1 to 1 or many to 1 relationship, instead they are a 1 to many relationships and we don't typically like this because we are always looking for an exact precise result due to our hubris and nature of wanting to predict things. And this kind of thinking and teaching really needs to stop. Why am I claiming this? Let's go back to when I previously mentioned rotating the vector <1,0> by 90 degrees we ended up at the point <0,1>, here we translated a vector by a 1/2 half step of a linear translation. A rotation by 90 degrees is a 1/2 step horizontal translation. If we translate by another 90 then we have a full linear translation along the same line. Here we can see that the unit vectors <1,0> and <0,1> both exist in the XY plane. We also know that by rotating a line around the unit circle doesn't break rotation at intervals of 90 + 180*n where n is an integer. If sin(90)/cos(90) = tan(90) is "undefined" then the clock on your wall wouldn't work and yet it does! How why? Well, we also know that when we multiply any value by i to give us a*i where a is any real value, this is the same thing as rotating a by 90 degrees in the Complex plane. Take the value of <5,0> rotate it by 90 degrees or multiply it by i and we will end up at <0, 5i>. If we look at the powers of i we will see this pattern: i^1 = sqrt(-1) = rotation by 90; i^2 = -1 = rotation by 180 degrees, i^3 = - sqrt(-1), rotation by 270 degrees and i^4 = 1 = rotation by either 0 or by 360 degrees and we've come full circle. And this is also has a modulus operational property. So, when we look at division by 0 and we look at it from within the context of slope, rotating by 90 degrees, multiplying by i, they are all equivalent. This is perpendicularity, well verticality within Perpendicularity to be more precise. Verticality is the reciprocal of Horizontality. Therefore, when we see a fraction or division by 0 in a/b where { a != 0; for all a } And b = 0 we can view these as the vectors of the form <0, a> where x is located at a. Within the context of slope, here we have 0 translation in X and only translation in Y. With this vector notation of x at a, for all a with no other translation in x but only in y. This is the vertical line at x = a and it is perpendicular to the point or value of <a,0>. Thus <a,0> and <0,a> are perpendicular values. The input value is a at x, and the output is all values of Y at x. This is the 1 to many relationship that most don't typically like. Here we have infinite slope, this is verticality. When we look at the relationship and the similarities between the sine and the cosine functions this again is self-evident. We know that they are continuous rotational, sinusoidal, circular, transcendental, periodic wave functions. We know that both of them have the same exact range and domain, their ranges are [-1,1] and their domains within the Reals is the set of all Reals. I won't directly get into complex composition here, but it can be inferred via the coordinate pair within the complex plane (cos(t), i*sin(t)). We also know that they have same period of 2PI or 360 degrees. We also know they have the same wave form; in other words, their shapes are the same. The only major differences between them are their initial starting positions, the properties that one is an ODD function, and the other is an EVEN function. The starting point for sine is <0,0> and the sine is an ODD function. The starting point for cosine is <0,1> and it is an even function. Their wave forms or their graphs are exactly 90-degree or PI/2 radian horizontal translations of each other. So here if we stop calling the "imaginaries" by that naming convention and instead call them what they are, the "orthogonals" with respect to the "reals" or better stated, the "perpendiculars" and if we start to treat ordinary values as vector entities, we can clearly see that every number that exists including 0, 1, all in between and all that extends out to infinity are Circular. The expression 1+1 = 2. Is the unit circle located with its center at the point (1,0). We can see this from the first operand being the vector <1,0> translating it horizontally in the +x direction by the vector <1,0> arriving at the new vector <2,0> with its center at <1,0> The sum or the addition of the two is the diameter of the circle. Division by 0 is not undefined. When the denominator is 0, it is perpendicular to its corresponding numerator in terms of a fraction, and in terms of division or by repeated subtraction, division by 0 the divisor is 0 and we are subtracting repeatedly by 0 to reduce the dividend to 0 and because of the additive identity property of a + 0 = a, this is the same as a - 0 = a, and there is no change to the dividend, and we can or would perform this subtraction an infinite amount of times never being able to reduce it. This is vertical slope or verticality within perpendicularity with respect to horizontality. So, in hindsight or in essence diving by 0 is almost equivalent to multiplying by i, or by taking a real value and rotating it by 90 degrees not from the X-axis to the Y-axis, but at x into the Y direction for all points in Y at x. This is Well defined! Sure, it might be ambiguous because of the generated infinites but it's not "undefined". Just food for thought. Think critically for yourself instead of being forced into believing something because that's what's written in the textbooks, and we must believe that. Yeah, and they are never always 100% right! They always have some error and or falsehoods either unintentionally or by design. Challenge everything, put it to the test!