sum_(k=1)^(oo)(-1)^(k)(3k-1)/(2k+1)

Check Convergence: The given series is an alternating series of the form ∑(k=1 to ∞) ((-1)^(k)(3k-1))/(2k+1). To find the sum of an infinite series, we need to determine if the series converges and, if it does, use a method suitable for finding the sum of an alternating series.Calculate Limit: First, we need to check if the series converges. An alternating series converges if the absolute value of the terms tends to zero as k tends to infinity and if the terms decrease in absolute value. Let's check the limit of the absolute value of the terms as k approaches infinity.Evaluate Limit: We calculate the limit of the absolute value of the terms: lim(k→∞) |(3k-1)/(2k+1)|. As k approaches infinity, the -1 and +1 become negligible, so the limit is essentially the limit of (3k)/(2k), which simplifies to 3/2. Since the limit is not zero, the series does not meet the criteria for convergence of an alternating series.

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Algebra 2

Sum of finite series starts from 1

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\sum_{k=1}^{\infty}(-1)^{k} \frac{3 k-1}{2 k+1}

Check Convergence: The given series is an alternating series of the form $\sum_{k=1}^{\infty} \frac{(-1)^k(3k-1)}{2k+1}$ . To find the sum of an infinite series, we need to determine if the series converges and, if it does, use a method suitable for finding the sum of an alternating series.
Calculate Limit: First, we need to check if the series converges. An alternating series converges if the absolute value of the terms tends to zero as $k$ tends to infinity and if the terms decrease in absolute value. Let's check the limit of the absolute value of the terms as $k$ approaches infinity.
Evaluate Limit: We calculate the limit of the absolute value of the terms: $\lim_{k\to\infty} \left|\frac{3k-1}{2k+1}\right|$ . As $k$ approaches infinity, the $-1$ and $+1$ become negligible, so the limit is essentially the limit of $\frac{3k}{2k}$ , which simplifies to $\frac{3}{2}$ . Since the limit is not zero, the series does not meet the criteria for convergence of an alternating series.