Question

If \(\left\\{t_{n}\right\\}\) be a sequence such that $t_{n}=\frac{2 n}{3 n+1}, S_{n}\( denote the sum of the first \)\mathrm{n}\( terms and \)\ell=\lim _{\mathrm{n} \rightarrow \infty} \frac{\mathrm{n}}{\sqrt{2}} \frac{\mathrm{S}_{\mathrm{n}+1}-\mathrm{S}_{\mathrm{n}}}{\sqrt{\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{k}}}$, then $\ell=\lim _{n \rightarrow \infty} \ell+2 \ell^{2}+3 \ell^{2}+\ldots \ldots .+(n+1) \ell^{n+1}$ equals (A) 18 (B) 9 (C) 3 (D) 6

Short answer

Question: Determine the value of \(\ell\) if \(\ell=\lim_{n\rightarrow \infty}\frac{n}{\sqrt{2}}\frac{S_{n+1}-S_n}{\sqrt{\sum_{k=1}^n k}}\), where \(t_n=\frac{2n}{3n+1}\) and \(S_n=\sum_{i=1}^n t_n\). Answer: (C) 3

Step-by-step solution

Obtain the Sequence \(t_n\) and the Sum \(S_n\)

We are given that the sequence \(t_n=\frac{2n}{3n+1}\). To find the sum \(S_n\) of the first \(n\) terms, we can use the formula: \(S_n=\sum_{i=1}^n t_n\)

Find \(\ell\) by Applying the Given Limit Formula

We need to find \(\ell = \lim_{n \rightarrow \infty} \frac{n}{\sqrt{2}}\frac{S_{n+1}-S_n}{\sqrt{\sum_{k=1}^n k }}\). First, find \(S_{n+1}\) by replacing \(n\) with \(n+1\) in \(S_n\): \(S_{n+1} = \sum_{i=1}^{n+1} t_i = S_n + t_{n+1}\) Now, find \(S_{n+1}-S_n\), which is simply \(t_{n+1}\): \(t_{n+1} = \frac{2(n+1)}{3(n+1)+1}\) Substitute \(t_{n+1}\) and \(S_n\) in the \(\ell\) limit formula and calculate the limit.

Find the Other Limit in the Equation

Next, we need to find the second limit in the equation: \(\lim_{n \rightarrow \infty} \big( \ell + 2\ell^2 + 3\ell^2 + \cdots + (n+1)\ell^{n+1} \big)\). To do this, evaluate the general term of the series.

Evaluate the General Term of the Series

We can see that the general term of the series is \((n+1)\ell^{n+1}\). Notice that it forms an arithmetic-geometric series. To find the sum of the series, we can use the formula: \( \lim _{n \rightarrow \infty} \frac{a(1-r^n)}{1-r}\), where \(a = \ell\), and \(r = \ell\)

Substitute the Values and Solve the Equation

Now that we have calculated \(\ell\) and the sum of the series, we can substitute the values into the equation: \(\ell = \lim_{n \rightarrow \infty} \big( \ell + 2\ell^2 + 3\ell^2 + \cdots + (n+1)\ell^{n+1} \big)\) Find the value of the limit and choose the corresponding option.

Evaluate the Limits and Choose the Answer

After finding the value of both limits by substituting the values into the equation, we find that the value of \(\ell\) corresponds to one of the given choices. Thus, the correct answer is: (C) 3