Evaluatesum_(n-1)^(oo)(2+n)/(1-2n)

Identify Series Form: The given series is an infinite series of the form sum_(n=1)^(∞)(2+n)/(1-2n). To evaluate this, we need to determine if the series converges and, if so, find its sum. We can start by looking for a pattern or a formula that represents the general term of the series.General Term Calculation: The general term of the series is a_n = (2+n)/(1-2n). To check for convergence, we can use the nth-term test for divergence, which states that if the limit of a_n as n approaches infinity is not zero, then the series diverges.Convergence Test: Let's calculate the limit of a_n as n approaches infinity: lim_(n→∞)(2+n)/(1-2n) = lim_(n→∞)(2/n + 1)/(1/n - 2). As n approaches infinity, the terms 2/n and 1/n approach 0, so the limit simplifies to: lim_(n→∞)(0 + 1)/(0 - 2) = -1/2.Limit Calculation: Since the limit of the general term a_n as n approaches infinity is -1/2, which is not equal to zero, the nth-term test for divergence tells us that the series does not converge. Therefore, the sum of the series is not a finite number.

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Math Problems

Algebra 2

Sum of finite series starts from 1

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Q. Evaluate

\sum_{n=1}^{\infty}\frac{2+n}{1-2n}

Identify Series Form: The given series is an infinite series of the form $\sum_{n=1}^{\infty}\frac{2+n}{1-2n}$ . To evaluate this, we need to determine if the series converges and, if so, find its sum. We can start by looking for a pattern or a formula that represents the general term of the series.
General Term Calculation: The general term of the series is $a_n = \frac{2+n}{1-2n}$ . To check for convergence, we can use the nth-term test for divergence, which states that if the limit of $a_n$ as $n$ approaches infinity is not zero, then the series diverges.
Convergence Test: Let's calculate the limit of $a_n$ as $n$ approaches infinity: $\newline$ $\lim_{n\to\infty}\frac{2+n}{1-2n} = \lim_{n\to\infty}\frac{\frac{2}{n} + 1}{\frac{1}{n} - 2}.$ $\newline$ As $n$ approaches infinity, the terms $\frac{2}{n}$ and $\frac{1}{n}$ approach $0$ , so the limit simplifies to: $\newline$ $\lim_{n\to\infty}\frac{0 + 1}{0 - 2} = -\frac{1}{2}.$
Limit Calculation: Since the limit of the general term $a_n$ as $n$ approaches infinity is $-\frac{1}{2}$ , which is not equal to zero, the nth-term test for divergence tells us that the series does not converge. Therefore, the sum of the series is not a finite number.