Taylor Series & Maclaurin Series Examples | Calculus 2 - JK Math

Example Problems For How to Represent Functions as Taylor & Maclaurin Series (Calculus 2)

In this video we look at practice problems of representing functions as Taylor series and Maclaurin series. This includes creating Taylor series from scratch, and using common Maclaurin series as a shortcut.

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This video series is designed to help students understand the concepts of Calculus 2 at a grounded level. No long, boring, and unnecessary explanations, just what you need to know at a reasonable and digestible pace, with the goal of each video being shorter than the average school lecture!

Calculus 2 requires a solid understanding of calculus 1, precalculus, and algebra concepts and techniques. This includes limits, differentiation, basic integration, factoring, equation manipulation, trigonometric functions, logarithms, graphing, and much more. If you are not familiar with these prerequisite topics, be sure to learn them first!

Video Chapters:
0:00 Example 1 - ln(x) centered at c=1
13:21 Example 2 - e^(3x) centered at c=2
19:48 Review of Common Maclaurin Series
22:19 Example 3 - x*cos(4x^2) as a Maclaurin series
26:42 Example 4 - ln(1+x^3) as a Maclaurin series
31:36 Outro

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[JKMath]

CORRECTION: In the first example of this video I make a sight error. The solution is almost correct, but needs a small adjustment. Here's what I should have done differently:

When finding f^(n) by finding a pattern in the derivatives of f(x), notice that the original function f(x) does not follow the same pattern as its derivatives do. It doesn't match (x)^(-n)*(-1)^(n+1)*(n-1)! which is what I said f^(n) was. The original function is just ln(x), which would represent the term of the series where n=0, and that does not follow the pattern the rest of the terms follow. So, what I should have done (and forgot to do in the video) is EXCLUDE the n=0 term, and start the series at n=1. So, the correct answer to the example would be the same power series I get in the video, except n should start at 1 not 0. (Note that this change will not affect the ratio test / interval of convergence, the I.O.C. would still be (0, 2] as I give in the video)

As a side note, notice that f(c) would be f(1)=ln(1)=0, and so that "first term" for n=0 would just be 0, so excluding it from the power series makes sense. You might not be able to tell this from the power series I find in the example, because if you were to try to find the n=0 term, the denominator would become 0 since the denominator is just n, and that immediately becomes an issue since you can't divide by 0. So really, that should have been an immediate tell that my answer needed a slight adjustment to be correct, but unfortunately I missed that error when recording & editing this video. 

Also, aside from just starting the series at n=1 to fix my mistake, another way that the power series could be fixed (while leaving the series to start at n=0) is to simply replace each "n" in the series with "n+1" and changing (-1)^(n+1) to (-1)^n. This would achieve the same result as starting the series at n=1. So another correct answer could be ∑^∞_(n=0) [(-1)^n(x-1)^(n+1)]/(n+1).

My sincerest apologies on this mistake! I hope this comment clears up any confusion, and if you have any further questions please reply to this comment and let me know!

-Josh

[cielonicolosi]

this video deserves way more likes your the first video for taylor series i've watched that actually makes me understand it thank you!