True-False Quiz Question 19: Proof that 0.99999... = 1 (Convergent Geometric Series) @mes

Math Easy Solutions | True-False Quiz Question 19: Proof that 0.99999... = 1 (Convergent Geometric Series) @mes | Uploaded February 2024 | Updated October 2024, 6 hours ago.
In this video I show that the number 0.99999.... can be written as a convergent geometric series whose sum is equal to 1. The number 0.99999... can be written as 0.9 + 0.09 + 0.009 + ... which can further be written as 0.9 + 0.9(0.1) + 0.9(0.1)^2 + ... which is just a geometric series. In our case, r = 0.1 and is less than 1, thus the geometric series is convergent and we can apply the formula for the sum, which is just 1. Thus the answer to the question is True: 0.99999 ... = 1. Fascinating stuff!

Time stamps:

- Question 19: 0:00
- Solution: True: Geometric Series 0:14
- Sum of a Convergent Geometric Series 3:04
- Sum of Geometric Series = 1: 5:00

Full video below:

- Infinite Sequences and Series: Review and True-False Quiz: youtu.be/F0dsQLdXXpI
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$Exercise 2: Deriving the 1st Order and 3rd Order Optics Formulas In this video, I derive the first-order and third-order optics formulas by approximating the refraction formula for light using first and third-order Taylor polynomials, respectively. Refraction refers to the change in the direction of light as it passes between different mediums due to a change in speed. Light naturally travels in a way that minimizes time, and this principle helps us derive formulas for how light refracts when entering a material. The distances light travels before and after passing through an optical lens can be calculated using the Law of Cosines, which leads to a complex equation. However, we simplify this equation using Taylor polynomial approximations. In the first part of the video, I derive the first-order, or Gaussian optics, formula by approximating the cosine function in the distance equations with its first term, which is simply 1. In the second part, I derive the third-order Gaussian optics formula by using the third-degree Taylor polynomial for cosine, which includes the first two non-zero terms. I also apply a given hint to approximate the inverse of the distances using the first two terms of their respective binomial series. Additionally, I use a second hint to approximate the angle with its sine, which introduces a term involving the height. After working through the algebra, I arrive at the third-order optics formula! Timestamps: - Exercise 2: Derive 1st Order and 3rd Order Optics Equations for refraction: 0:00 - Solution to (a): Approximating cosine with First degree Taylor Polynomial: 0:59 - Equation 3: Gaussian optics or 1st order formula: 8:26 - Solution to (b): Approximating cosine with Third degree Taylor Polynomial: 9:19 - Modifying lengths in form of Binomial series: 17:46 - 1st Hint: Approximate lengths using first 2 terms of Binomial series: 23:20 - Plugging in our approximations into Equation 1: 28:06 - 2nd Hint: Approximate ɸ with sin(ɸ) to include the height term: 43:48 - Equation 4: 3rd order optics formula: 46:37 Full video, notes, and playlists: - Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0F76sIU8xm09oqBTq1mlry3 - HIVE notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-applications-of-taylor-polynomials - Infinite Sequences and Series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz . Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps$