Representing functions as power series, Single Variable Calculus

(Previous video: https://youtu.be/6pfNKkWUWNk) We do five examples of modifying the prototype 1/(1-x) = Σ x^n to create power series representations for new functions. We also specify the interval and radius of convergence for each. In mathematics, particularly in the field of analysis, the representation of a function as a power series involves expressing involves the function 𝑓(𝑥) as a power series 𝑓(𝑥)=∑_{𝑛=0}^∞ 𝑐_𝑛(𝑥−𝑎)^𝑛, for any 𝑥 values in the interval of convergence. This representation is fundamental in various mathematical areas, including calculus, differential equations, and numerical analysis, as it simplifies analyzing difficult functions into analyzing infinite polynomial-like forms that are often easier to manipulate and study. This lesson focuses on finding power series representations for various functions and determining their intervals and radii of convergence. We approach each function in this video by rearranging it into a form resembling the prototype 1/(1−𝑥) and then substituting the power series representation. We deduce what the convergence properties are based on the transformed series. Example 1: 𝑓(𝑥)=𝑥/(1−𝑥). We rearrange this function as 𝑓(𝑥)=𝑥⋅1/(1−𝑥) and use the power series representation of 1/(1−𝑥), which is ∑_{𝑛=0}^∞ 𝑥^𝑛. Therefore, the power series for 𝑓(𝑥) is: 𝑓(𝑥) = 𝑥⋅∑_{𝑛=0}^∞ 𝑥^𝑛 = ∑_{𝑛=0}^∞ 𝑥^{𝑛+1}. The interval of convergence is inherited from the original series: 𝐼=(−1,1), with a radius of convergence 𝑅=1. Example 2: 𝑓(𝑥)=𝑥/1+𝑥. We rewrite this as 𝑥⋅1/(1−(−𝑥)) and substitute the power series for 1/(1−(−𝑥)), getting: 𝑓(𝑥) = 𝑥⋅∑_{𝑛=0}^∞ (−𝑥)^𝑛 = ∑_{𝑛=0}^∞ (−1)^𝑛 𝑥^{𝑛+1}. This series converges for |𝑥| less than 1, so the interval of convergence is (−1,1) with a radius of 1. Example 3: 𝑓(𝑥)=1/1−(5𝑥^2). We directly use the power series representation: 𝑓(𝑥) = ∑_{𝑛=0}^∞ (5𝑥^2)^𝑛 = ∑_{𝑛=0}^∞ 5^𝑛 𝑥^{2𝑛} The interval of convergence is determined by |5𝑥^2| less than 1. Thus, the interval of convergence is (−1√5,1√5) with a radius of 1√5. Example 4: 𝑓(𝑥) = 𝑥^3/(𝑥+10). Example 5: 𝑓(𝑥) = 7𝑥/(2^5−𝑥^5). Check how we do these in the video! Thanks for watching. #mathematics #math #calculus #sequencesandseries #powerseries #iitjammathematics #taylorseries #maclaurinseries #mathtutorial #seriesconvergence