Determinate the convergence or divergence of $\sum _{n=0}^{\mathrm{\infty}}\frac{(-1{)}^{n}}{\sqrt{n+1}}$

Jaylene Hunter
2022-07-17
Answered

Determinate the convergence or divergence of $\sum _{n=0}^{\mathrm{\infty}}\frac{(-1{)}^{n}}{\sqrt{n+1}}$

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Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.

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asked 2022-06-01

Consider the sequence

${a}_{n}=\sqrt[n]{n\cdot {2}^{3n}+{3}^{2n}}$

With a string of inequalities, one can show that ${a}_{n}$ is bounded and the graph of the function $f(x)=\sqrt[x]{x\cdot {2}^{3x}+{3}^{2x}}$ suggests that f is monotone, but how could one prove convergence and a calculate the limit of ${a}_{n}$ ?

${a}_{n}=\sqrt[n]{n\cdot {2}^{3n}+{3}^{2n}}$

With a string of inequalities, one can show that ${a}_{n}$ is bounded and the graph of the function $f(x)=\sqrt[x]{x\cdot {2}^{3x}+{3}^{2x}}$ suggests that f is monotone, but how could one prove convergence and a calculate the limit of ${a}_{n}$ ?