What are Taylor Series? Part 1

Taylor's Theorem is discussed and a proof is given using only elementary calculus methods based on the mean value theorem for integrals. The approach is simple and straightforward and meant to be accessible.

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Education

Tags:

• Taylor's
• Theorem
• Taylor
• Series
• Calculus
• Analysis
• Mathematics
• Math
• Applied

[Mathview]

Taylor's theorem examines the convergence of the Taylor series and makes it possible for us to use such series correctly. MacLauren seems to have developed what we call formal series expansion, that is without proof of their convergence. Personally, I give Taylor the credit and call them all Taylor series.

[MrAsiansunite]

@Mathview Noone call it Macloren series, but Macloren formula.... And the issue is not about convergence but about Lagrange, Couchy and Peano remainings. Macloren formula gives a good valuation of the remainings, ok? Study more deeply and them post a video

[Mathview]

@MrAsiansunite Thank you for your comment. I believe the history is that MacLauren came before Taylor, and formally the MacLauren series is a special case of the Taylor series. Therefore All MacLauren series are Taylor series, but not all Taylor series are MacLauren series. The central feature of the Taylor Theorem with remainder, which we discuss here, is the Lagrange Remainder Term which gives us a tool to examine and verify the convergence of formal infinite series.

[Mathview]

@MrAsiansunite A simple problem illustratates the difference between formal Taylor and Maclauren series. PROBLEM: Find the Maclauren series for the function f(x) = 1/x .

[ppanayotov91]

Isn't what you do in the Example in the beginning (finding the coefficients by calculating the derivative at x=0) in fact a Maclaurin series? A Taylor series is around a number "a", not specifically 0...

[Mathview]

@ppanayotov91 One must confirm the convergence of the resulting series by evaluating the remainder. The remainder term then contains of (x-a)^n+1. I wanted to keep the presentation simple as possible here and omitted this point. In short, the Maclauren series is special case of the general Taylor series. Good comment!

[1000NESChip]

@MrAsiansunite

The series are referred to as MacLaurin "Series", not as Macloren "formula". Study Calculus and Engrish more deeply and "them" post a comment, OK? Your point is splitting hairs and trivial. Why don't you post a video helping others if you know so much? Crawl back into your smelly hole, you pretentious lowlife.

[Minnesota213484]

this guy is horrible, fucking 8 second pauses in between every word! come on dude lets go!!!

[MrPostalMatt]

Fuarkkk. Note to self, do not read the comments since they will only confuse you more lol. I'm such a noob in cal2.