I am sorry i still don't know how to start. and i am sorry for the problem earlier..it wasn't my fault :(
Note that the way students like you are asked to solve this and similar problems is usually not the way people like me solve such problems.
It is a good exercise to plug in the function in the formula for Taylor expansions and do the differentiations and try to obtain the general formula for the n-the term.
However, such derivations can be tedious and it's easy to make mistakes. That's why sooner or later you will be required to know the series expansions of the standard functions like sin(x) cos(x) exp(x) etc. Then you can quickly derive series expansions without repeatedly differentiating functions.
Also, note that computer algebra systems do not use Taylor's formula to derive Taylor series at all! They utilize the expansions for standard functions plus some clever tricks.
In fact, if you ask a comuter algebra system to find the n-th derivative at some point, it will actually derive the n-th order term in the Taylor expansion around that point and extract the derivative that way. The number of computations required for that are of order Log(n). This means that computing the millionth derivative of some complicated function at some given point requires no more than a few dozen of operations.
I don't think anything happened, it does to some accounts. how can I help.
Hey bot can you help?
It's alright, there's no need to apologize. Starting with something new can sometimes be challenging, but I'm here to help you. To begin understanding the process, let's break it down into steps:
Step 1: Understand the concept of Taylor series expansion.
A Taylor series expansion is a way to represent a function as an infinite sum of terms. Each term in the expansion represents the contribution from a specific derivative of the function evaluated at a specific point.
Step 2: Determine the function you want to find the Taylor series expansion for.
In this case, you mentioned sin(x) as the function for which you want to find the Taylor series expansion.
Step 3: Understand the formula for Taylor series expansion.
The formula for the Taylor series expansion of a function f(x) around a point a is as follows:
f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
Step 4: Derive the derivatives.
To calculate each term in the series, you'll need to find the derivatives of the function. For sin(x), the derivatives are:
f(x) = sin(x)
f'(x) = cos(x)
f''(x) = -sin(x)
f'''(x) = -cos(x)
Step 5: Evaluate the derivatives at the point a.
In the formula above, you'll see that each term is multiplied by the value of the derivative evaluated at the point a. For example, if a is 0, then f(a) would be sin(0) = 0, f'(a) would be cos(0) = 1, f''(a) would be -sin(0) = 0, and so on.
Step 6: Simplify the terms.
After evaluating each derivative at the point a, plug the values into the formula and simplify the terms accordingly. Note that the terms involve powers of (x-a) divided by factorials.
Step 7: Combine the terms.
Once you have simplified each term, sum them up to get the Taylor series expansion.
Step 8: Understand the concept of convergence.
Remember that a Taylor series expansion is an infinite sum. This means that it may not converge for all values of x. You'll need to consider the convergence properties of the series to determine the range of x values for which the approximation is valid.
These steps should give you a general idea of how to start finding the Taylor series expansion for a given function. Remember that practice is key, and don't hesitate to ask for further clarification or assistance.