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- Understand behavior and pattern: To solve the infinite series , we first need to understand the behavior of the series and look for a pattern or a telescoping property that simplifies the summation.
- Rewrite terms as fractions: We notice that the terms inside the logarithm can be rewritten as a fraction of two consecutive terms of a sequence. Let's consider the sequence . Then, we can express the terms inside the logarithm as .
- Explore telescoping pattern: Now, let's write out the first few terms of the series to see if there is a telescoping pattern:\ln\left(\frac{a_1}{\(2\)a_1\(-1\)}\right) + \ln\left(\frac{a_2}{\(2\)a_2\(-1\)}\right) + \ln\left(\frac{a_3}{\(2\)a_3\(-1\)}\right) + \ldots\(\newline= \ln\left(\frac{}{\cdot }\right) + \ln\left(\frac{}{\cdot }\right) + \ln\left(\frac{}{\cdot }\right) + \ldots= \ln\left(\frac{}{}\right) + \ln\left(\frac{}{}\right) + \ln\left(\frac{}{}\right) + \ldots
- Expand terms using properties: We can see that each term can be rewritten using the properties of logarithms as . This will allow us to see if the series telescopes when we expand the terms.
- Consider differences between terms: Let's expand the first few terms using this property:
- Express terms using differences: We notice that there is no immediate cancellation between the terms. However, we can try to rewrite the series by pairing terms differently to see if a telescoping pattern emerges. Let's consider the difference between consecutive terms of the sequence :
- Check convergence of the series: Now, let's express the terms of the series using the differences :
- Calculate limit as approaches infinity: We can see that the terms and do not cancel out with adjacent terms in the series. This means that the series does not telescope in a straightforward way. We need to find another approach to evaluate the sum.
- Determine series convergence: Since the series does not telescope, we can consider the convergence of the series. We know that for a series to converge, the limit of as approaches infinity must be zero. Let's check the limit of as approaches infinity.
- Determine series convergence: Since the series does not telescope, we can consider the convergence of the series. We know that for a series to converge, the limit of as approaches infinity must be zero. Let's check the limit of as approaches infinity.Taking the limit, we have:
- Determine series convergence: Since the series does not telescope, we can consider the convergence of the series. We know that for a series to converge, the limit of as approaches infinity must be zero. Let's check the limit of as approaches infinity.Taking the limit, we have:= = Since the limit of the terms as approaches infinity is , which is a constant, the terms of the series do not approach zero. This means that the series does not converge, and therefore, it does not have a finite sum.
