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Geometric Sequenceshard
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Let ana_nan​ be a geometric sequence with first term a1>0a_1 > 0a1​>0 and common ratio 0<r<10 < r < 10<r<1. Define two functions for values of xxx where the series converge: f(x)=∑n=1∞anxnandg(x)=∑n=1∞(−1)n−1anxnf(x) = \sum_{n=1}^{\infty} a_n x^n \quad \text{and} \quad g(x) = \sum_{n=1}^{\infty} (-1)^{n-1} a_n x^nf(x)=∑n=1∞​an​xnandg(x)=∑n=1∞​(−1)n−1an​xn If f(1)=2f(1) = 2f(1)=2 and g(1)=23g(1) = \frac{2}{3}g(1)=32​, which of the following statements is/are correct?