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I am sorry i still don't know how to start. and i am sorry for the problem 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.

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