I mean, you can simply recognize that the series the expansion via either polynomial division or general binomial theorem has the generating function (1-x)^-2 thus the result is 1/(1-x)
First method: It gets somewhat easier if we don't forget the constant of integration but set it to 1 + C, so Intgr S dx = x + x² + x³ + ... + 1 + C = 1 + x + x² + x³ + ... + C = 1 / (1 - x) + C. So differentiation gets easier with no x in the nominator and C vanishes anyway.
@@bsmith6276 ... Actually it's a great problem because it wasn't a Geometric series problem although it had an |r| not equal to 0. The concept of 1 + 2x + 3x^2 + 4x^3 + ... = S was not implying a radius r where |r| < 1 he was stating. But a great two methods RUclips video watch getting back to my earlier statement and watching a new problem not Gauss's Summation Law nor Geometric Series Law anyone can find in any Algebra II High School textbook. I would have loved more chapter concepts when I was a Junior in High School!
@@bsmith6276 ...How many times do you have to see the summation of 1 + x + x^2 + x^3 + x^4 + ... with |x| < 1 Geometric sum before ... read my first comment again. Also if this is a new concept you haven't seen in Agebra II (second Algebra High School Algebra year in academics) I have to question where you are going to school before a university or college have as teachers. It turned out watching this full video the problem is from the same High School Algebra II textbook in the Gauss' sum Law of series chapter (probably the section on Geometric series in x).
Another great video! I really like the way you usually show more than one method for solving problems.
Good question and good reply
Brilliant as usual
I mean, you can simply recognize that the series the expansion via either polynomial division or general binomial theorem has the generating function (1-x)^-2 thus the result is 1/(1-x)
ez, i instantly recognized with generating functions
What about infinite polynomila with finite solutions?
Small correction that 1-x must be in mod i.e abs func.
First method: It gets somewhat easier if we don't forget the constant of integration but set it to 1 + C, so Intgr S dx = x + x² + x³ + ... + 1 + C = 1 + x + x² + x³ + ... + C = 1 / (1 - x) + C.
So differentiation gets easier with no x in the nominator and C vanishes anyway.
It's a Geometric series of |x|< 1 of k/(1 - r) following Gauss's summation Law of a sequential series. Too easy! 😂🤣
Not every problem on a video has to be an olympiad level problem.
@@bsmith6276 ... Actually it's a great problem because it wasn't a Geometric series problem although it had an |r| not equal to 0. The concept of 1 + 2x + 3x^2 + 4x^3 + ... = S was not implying a radius r where |r| < 1 he was stating. But a great two methods RUclips video watch getting back to my earlier statement and watching a new problem not Gauss's Summation Law nor Geometric Series Law anyone can find in any Algebra II High School textbook. I would have loved more chapter concepts when I was a Junior in High School!
@@lawrencejelsma8118 So if this is actually a good problem then what is with your original "too easy" complaint with the crying emojis?
@@bsmith6276 ...How many times do you have to see the summation of 1 + x + x^2 + x^3 + x^4 + ... with |x| < 1 Geometric sum before ... read my first comment again. Also if this is a new concept you haven't seen in Agebra II (second Algebra High School Algebra year in academics) I have to question where you are going to school before a university or college have as teachers. It turned out watching this full video the problem is from the same High School Algebra II textbook in the Gauss' sum Law of series chapter (probably the section on Geometric series in x).