dunkelheit
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- Sep 7, 2018
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[MATH]\lim_{n \to \infty} \frac{n}{n+2} \sum_{k=1}^n \frac{k}{k+3}[/MATH]My attempt: since [MATH]\frac{k}{k+3}[/MATH] is increasing for [MATH]k \geq 1[/MATH] we have that [MATH]\frac{k}{k+3} \geq \frac{1}{4}[/MATH] for all [MATH]k \geq 1[/MATH]; since it holds for all [MATH]k \geq 1[/MATH] it holds if we sum up to [MATH]n[/MATH], hence
[MATH]\sum_{k=1}^n \frac{k}{k+3} \geq \sum_{k=1}^n \frac{1}{4} = \frac{n}{4}[/MATH]Since [MATH]n\in\mathbb{N}[/MATH] it is [MATH]\frac{n}{n+2} \geq 0[/MATH] so we can multiply both sides of the inequality by [MATH]\frac{n}{n+2}[/MATH] and not changing the verse of the inequality, so
[MATH]\frac{n}{n+2}\sum_{k=1}^n \frac{k}{k+3} \geq \frac{n}{4}\frac{n}{n+2}[/MATH]Letting [MATH]n \to \infty[/MATH] both sides we have
[MATH]\lim_{n \to \infty} \frac{n}{n+2}\sum_{k=1}^n \frac{k}{k+3} \geq \lim_{n \to \infty}\frac{n}{4}\frac{n}{n+2}=\infty[/MATH]So by comparison
[MATH]\lim_{n \to \infty} \frac{n}{n+2} \sum_{k=1}^n \frac{k}{k+3}=\infty[/MATH]Is this correct? Any advice or suggestion for improving is welcome, thanks.
[MATH]\lim_{n \to \infty} \frac{n}{n+2} \sum_{k=1}^n \frac{k}{k+3}[/MATH]My attempt: since [MATH]\frac{k}{k+3}[/MATH] is increasing for [MATH]k \geq 1[/MATH] we have that [MATH]\frac{k}{k+3} \geq \frac{1}{4}[/MATH] for all [MATH]k \geq 1[/MATH]; since it holds for all [MATH]k \geq 1[/MATH] it holds if we sum up to [MATH]n[/MATH], hence
[MATH]\sum_{k=1}^n \frac{k}{k+3} \geq \sum_{k=1}^n \frac{1}{4} = \frac{n}{4}[/MATH]Since [MATH]n\in\mathbb{N}[/MATH] it is [MATH]\frac{n}{n+2} \geq 0[/MATH] so we can multiply both sides of the inequality by [MATH]\frac{n}{n+2}[/MATH] and not changing the verse of the inequality, so
[MATH]\frac{n}{n+2}\sum_{k=1}^n \frac{k}{k+3} \geq \frac{n}{4}\frac{n}{n+2}[/MATH]Letting [MATH]n \to \infty[/MATH] both sides we have
[MATH]\lim_{n \to \infty} \frac{n}{n+2}\sum_{k=1}^n \frac{k}{k+3} \geq \lim_{n \to \infty}\frac{n}{4}\frac{n}{n+2}=\infty[/MATH]So by comparison
[MATH]\lim_{n \to \infty} \frac{n}{n+2} \sum_{k=1}^n \frac{k}{k+3}=\infty[/MATH]Is this correct? Any advice or suggestion for improving is welcome, thanks.
