Proof of 0.9999…. = 1 using geometry series.

#math
#series
#algebra

[DrYapKeemSiah]

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Following are some of my videos on calculus, algebra, logarithm and trigonometry:

Calculus/Integration
[1] Solving Gaussian Integral (integration of gaussian function) by Feynman’s Method.
[2] Created another 1 minute video for Happy π day! Solve an integral and the answer is π!
[3] Solving integral x^(9)*(x^(10) + 1)/(x^(20) + 1) dx. Can we solve it using substitution?
[4] Solving integral 1/(-sqrt(x^2 + 4*x + 3)) dx. Can it be solved by substitution?
[5] Solving sum of k=0 to ∞ for integral sin^(2k) (x) dx. Solution is much easier than expected!
[6] Solving area of region bounded between 2 curves, i.e., {(x, y): x^2 ≤ y ≤ x + 2} using integration.
[7] Solving integral (x^2 - 1)/(x^4 - 1) dx.
[8] Given y=kx^2, x=ky^2, k is +ve constant & area enclosed between parabola curves is 1 unit^2. Find k.
[9] Given f(x) = integral (x^5-16x)/(x^8-256) dx and f(-2) = π/32. Find f(2). Explanation in 60 seconds

Algebra
[1] Is this a quartic or quadratic problem? Can we solve it within 60 seconds and without calculator?
[2] Solving (10016*104*12*8 + 256)^0.5 but without using calculator & computer. Answer is a nice number!
[4] 15 seconds video on solving sqrt(10300 - 9*10*10*11) without using calculator and computer.
[5] Given x^2 - x - 1 = 0, solve cos^1 ((3/2)*(7*x + 1)/(x^9 - 18*x - 13)).
[6] An infinite radical equation, i.e., sqrt(3*sqrt(7*sqrt(…))) solve by Geometric Series.
[7] If (√ 2 + √ 3)^11 = a √ 2 + b√ 3, then (a² - (3/2)*b²)^11 = ?
[8] Given (|x-3| - 4)^2 = 4. How many answers for x?
[9] Solving an interesting infinite power equation.

Logarithm/exponential
[1] Solving a finite power equation of (ln(x))^2. The solution is surprisingly easier than expected.
[2] Given that ln(3) is approximately equal to 1.099, find ln(3.1) using calculus without calculator.
[3] Solving x^(ln x) = x for real numbers. How many answers for x?
[4] Solving a radical-absolute equation. Is this a quadratic equation too?
[5] Solving a logarithm-quadratic question of Stanford Math Tournament 2021. Calculator is not allowed.
[6] 15 seconds video to explain the answer for ln(-1) based on principal value of trigonometry.
[7] 15 seconds video on solving x^(ln|x|) = 1/|x|. There are 4 answers for x?

Trigonometry
[1] Solving a trigonometry-power equation? Solution is fairly simple and can be explained in 60 seconds.
[2] 45 seconds video deriving cos(-θ)=cos(θ) & sin(-θ)=-sin(θ). Basic identities of trigonometry Part 5.
[3] 15 seconds video for deriving sec^2(x) = tan^2(x) + 1. Basic identities of trigonometry Part 1.
[4] 15 seconds video for deriving sin^2(x) + cos^2(x) = 1. Basic identities of trigonometry Part 2.

[fritz]

Isn't any number raised to infinity is equal to infinity? so 0.9(1-0.1 raised to infinity)/1-0.1 = negative infinity.