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Write the first four non-zero terms of the power series for about
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Q. Write the first four non-zero terms of the power series for about
- Expand Cosine Taylor Series: To find the power series for about , we need to expand the cosine function into its Taylor series and then multiply by and subtract . The Taylor series for about is , where is the input to the cosine function. In our case, .
- Simplify Series Terms: First, we write down the Taylor series expansion for about :
- Multiply by : Now, we simplify the terms of the series:
- Simplify Coefficients: Next, we multiply each term by to get the series for :
- Subtract : We simplify the coefficients:
- Identify Non-Zero Term: Now, we subtract from the series to get the final series for :
- Final Power Series Terms: We notice that the constant term cancels with the constant term from when , so the first non-zero term is actually the term with :
- Final Power Series Terms: We notice that the constant term cancels with the constant term from when , so the first non-zero term is actually the term with :The first four non-zero terms of the power series for are: and
