QuestionUse properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the function given below. Use the Taylor series , for .
The first term is
Studdy Solution
STEP 1
1. We are given the Taylor series expansion for .
2. We need to find the Taylor series for by using substitution and properties of power series.
3. We are looking for the first four nonzero terms.
STEP 2
1. Substitute for in the given Taylor series.
2. Calculate the first four nonzero terms of the new series.
STEP 3
Substitute for in the given Taylor series .
This substitution transforms each in the original series to .
STEP 4
Calculate the first four nonzero terms:
1. The first term is .
2. The second term is .
3. The third term is .
4. The fourth term is .
Thus, the first four nonzero terms of the Taylor series for are:
The first term is .
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