sum_(n=1)^(oo)(n+1)/(nsqrtn)

Recognize Non-Standard Series: Recognize the series as a non-standard series that does not directly fit a common summation formula. We need to analyze the terms to determine if the series converges or diverges.Simplify Terms: Simplify the term (n+1)/(n√n) to (n+1)/n^(3/2). This can be further simplified to 1/n^(1/2) + 1/n^(3/2).Split into Two Series: Split the series into two separate series: sum_(n=1)^(oo)1/n^(1/2) and sum_(n=1)^(oo)1/n^(3/2).Determine Convergence: Determine the convergence or divergence of the first series sum_(n=1)^(oo)1/n^(1/2). This is a p-series with p = 1/2, which is less than 1. Therefore, by the p-series test, this series diverges.Final Result: Since the first series diverges, there is no need to test the second series. The sum of the entire series is undefined because the series does not converge.

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$\sum_{n=1}^{\infty} \frac{n+1}{n \sqrt{n}}$

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Algebra 2

Sum of finite series starts from 1

Full solution

\sum_{n=1}^{\infty} \frac{n+1}{n \sqrt{n}}

Recognize Non-Standard Series: Recognize the series as a non-standard series that does not directly fit a common summation formula. We need to analyze the terms to determine if the series converges or diverges.
Simplify Terms: Simplify the term $(n+1)/(n\sqrt{n})$ to $(n+1)/n^{3/2}$ . This can be further simplified to $1/n^{1/2} + 1/n^{3/2}$ .
Split into Two Series: Split the series into two separate series: $\sum_{n=1}^{\infty}\frac{1}{n^{1/2}}$ and $\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}$ .
Determine Convergence: Determine the convergence or divergence of the first series $\sum_{n=1}^{\infty}\frac{1}{n^{1/2}}$ . This is a p-series with $p = \frac{1}{2}$ , which is less than $1$ . Therefore, by the p-series test, this series diverges.
Final Result: Since the first series diverges, there is no need to test the second series. The sum of the entire series is undefined because the series does not converge.