HELP! I'm in calculus class and they keep talking about Taylor series. I don't understand what's going on. Please explain what is Taylor series and what is Mclaurin series and how they are similar and how they are different.

Essentially, the Taylor series is the default equation whereas the Mclaurin series is one in which the p can change. Whereas both are used to write nonzero terms, only one has different p values.

Essentially the two series serve the same purpose. The only difference between the two is that in Mclaurin series the center is always zero.

Taylor series and the associated McLauran series are + x^5

series of infinite terms allowing us to find some common mathematical functions
eg
sin(x) = x - x^3/3! + x^5/5! - x^7/7! + .... , x must be in radians
e^x = 1 + x + x^2/2! + x^3/3! + ....
ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...., -1 < x < 1

They must converge, even though some of them might diverge initially.

google Taylor series, there are some really good youtubes from the KhanAcademy

First sentence should say:

Taylor series and the associated McLauran series are series of infinite terms allowing us to find some common mathematical functions.

Both are the same idea

Taylor expands starting at some point where x = a
terms are something times (x-a)^n

Maclaurin assumes a = 0, terms are in x^n

thus Maclaurin is a special case of Taylor where we start at x = 0

Sure! I'd be happy to explain Taylor series and Maclaurin series in calculus.

A Taylor series is a mathematical representation of a function as an infinite sum of terms. It expresses a function as the sum of an infinite number of terms, where each term represents a certain derivative of the function evaluated at a specific point. The general form of a Taylor series for a function f(x) centered around a point a is:

f(x) = f(a) + f'(a)(x - a) + (f''(a)/2!)(x - a)^2 + (f'''(a)/3!)(x - a)^3 + ...

Essentially, a Taylor series allows us to approximate a function using its derivative values at a particular point. By including more and more terms in the series, we can obtain a more accurate representation of the original function.

A Maclaurin series is a specific type of Taylor series, where the series is centered around the point a = 0. In other words, a Maclaurin series is a Taylor series expansion where the point of expansion is the origin. This makes calculations simpler because it eliminates the need to evaluate derivatives at a specific point.

To find the coefficients of a Taylor or Maclaurin series, we need to find the corresponding derivatives of the function. The n-th derivative of a function f(x) can be found using various differentiation rules. By evaluating these derivatives at the center (a or 0), we can easily calculate the coefficients for the series.

To summarize the similarities and differences between Taylor and Maclaurin series:

Similarities:
1. Both Taylor and Maclaurin series can approximate a function using a sum of terms.
2. They both involve taking derivatives of the function.

Differences:
1. Taylor series is centered around a specific point (a), while Maclaurin series is centered around 0.
2. Maclaurin series is a specific type of Taylor series.
3. Calculating coefficients for a Taylor series involves evaluating derivatives at a specific point, whereas for a Maclaurin series, derivatives are evaluated at 0.

I hope this clears up your confusion about Taylor and Maclaurin series in calculus! Let me know if you have any further questions.