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For example, in calculating the following limit

$$\lim_{n\to\infty}\sum_{k=1}^{3n}\frac{1}{\sqrt{n^2+k}}$$

I know (because my professor wrote) that one of the methods of solving this limit is via the sandwich rule since we know the lower and upper bound

$$\frac{3n}{\sqrt{n^2+3n}}<\sum_{k=1}^{3n}\frac{1}{\sqrt{n^2+k}}<\frac{3n}{\sqrt{n^2+1}}$$

But I don't understand why it's so, because I would have guessed it would be

$$\frac{1}{\sqrt{n^2+3n}}<\sum_{k=1}^{3n}\frac{1}{\sqrt{n^2+k}}<\frac{1}{\sqrt{n^2+1}}$$ instead

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3 Answers 3

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Your guess is right in terms of bounding but you are forgetting about $\color{blue}{\text{summing}}$: $$\color{blue}{\sum_{k=1}^{3n}}\frac{\color{red}1}{\sqrt{n^2+3n}}<\sum_{k=1}^{3n}\frac{1}{\sqrt{n^2+k}}<\color{blue}{\sum_{k=1}^{3n}}\frac{\color{red}1}{\sqrt{n^2+1}} \\ \frac{\color{blue}{3n}}{\sqrt{n^2+3n}}<\sum_{k=1}^{3n}\frac{1}{\sqrt{n^2+k}}<\frac{\color{blue}{3n}}{\sqrt{n^2+1}}$$

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A single term of the series obeys the inequality $$\frac{1}{\sqrt{n^2 + 3n}} < \frac{1}{\sqrt{n^2 + k}} \le \frac{1}{\sqrt{n^2 + 1}}$$ for any integer $k \in \{1, 2, \ldots, 3n\}$, but the sum does not. Instead, you have to sum each of the expressions accordingly: $$\sum_{k=1}^{3n} \frac{1}{\sqrt{n^2 + 3n}} < \sum_{k=1}^{3n} \frac{1}{\sqrt{n^2 + k}} \le \sum_{k=1}^{3n} \frac{1}{\sqrt{n^2 + 1}},$$ and because the leftmost and rightmost sums are independent of $k$, we get

$$\frac{3n}{\sqrt{n^2 + 3n}} < \sum_{k=1}^{3n} \frac{1}{\sqrt{n^2 + k}} \le \frac{3n}{\sqrt{n^2 + 1}}.$$

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Since $k$ is in the denominator, the bigger $k$ the smaller the term of the sum. The biggest term of the sum is therefore $\frac 1 {\sqrt{n^2 + 1}}$. If you sum up this term $3n$ times, you get something that is definitely bigger than your original sum. But since this sum is independent of $k$, you can write it as a simple fraction with $3n$ in the numerator.

$$\sum_{k=1}^{3n}{\frac 1 {\sqrt{n^2 + k}}}<\sum_{k=1}^{3n}{\frac 1 {\sqrt{n^2 + 1}}}=\frac {3n} {\sqrt{n^2 + 1}}$$

Similarly, the smallest term of your sum will be $\frac 1 {\sqrt{n^2 + 3n}}$. Summing that up and simplifying you get:

$$\sum_{k=1}^{3n}{\frac 1 {\sqrt{n^2 + 3n}}}=\frac {3n} {\sqrt{n^2 + 3n}}<\sum_{k=1}^{3n}{\frac 1 {\sqrt{n^2 + k}}}$$

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