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- Understand Behavior of Cosine Function: We are given the infinite series . To solve this, we need to understand the behavior of the cosine function as its argument approaches zero. The Taylor series expansion of around is . When is very small, is approximately . Since becomes very small as increases, we can use this approximation for large .
- Analyzing Term for Large : Let's analyze the term for large . Using the approximation for small , we get . This approximation becomes more accurate as increases.
- Multiplying by : Now, we multiply the approximated value by as per the series definition. This gives us for large . However, we must be cautious because this approximation does not hold for small values of , specifically when is or .
- Calculating for and : For , the term is () = , which is a finite number. For , the term is () = , which is also a finite number. These terms do not follow the approximation and must be calculated separately.
- Explicit Calculation for First Two Terms: We calculate the first two terms explicitly. For , the term is . For , the term is also . So the sum of the first two terms is approximately .
- Using Approximation for : For , we can use the approximation for each term in the series. However, we must recognize that the series does not converge, as it is the sum of an infinite number of constant terms. This means the series diverges to negative infinity.
- Recognizing Divergence: Since the series diverges for , the sum of the entire series from to infinity also diverges. Therefore, the series does not have a finite sum.
