TIFR-2020-Maths-A: 1

Question

Consider the sequences $\left \{ a_{n}\right \}_{n=1}^{\infty }$ and $\left \{ b_{n}\right \}_{n=1}^{\infty }$ defined by

                                              $a_{n}=\left ( 2^{n}+3^{n} \right )^{1/n}$ and $b_{n}=\frac{n}{\sum_{i=1}^{n}\frac{1}{a_{i}}}$.

What is the limit of $\left \{ b_{n}\right \}_{n=1}^{\infty }$?

• A. $2$
• B. $3$
• C. $5$
• D. The limit does not exist

Answer 1

$\text{Here, sequence $b_n$ is the harmonic mean of terms in the sequence of}$ $a_n.$

$\text{Since, if a series converges to a number $L$ then its harmonic mean also converges to the same number}$ $L$.(Refer)

$\text{So, here, both sequences $a_n$ and $b_n$ converges to the same number. Hence, we have to find}$ $\lim_{n\rightarrow \infty}a_n = \lim_{n\rightarrow \infty} (2^n + 3^n)^{1/n}$

$y= \lim_{n\rightarrow \infty} (2^n + 3^n)^{1/n}$

$\ln y =\lim_{n\rightarrow \infty} \frac{1}{n} \ln (2^n + 3^n)$

$\ln y =\lim_{n\rightarrow \infty} \frac{1}{n} \ln (2^n(1+(\frac{3}{2})^n))$

$\ln y =\lim_{n\rightarrow \infty} \frac{1}{n} \ln (2^n)+ \frac{1}{n}\ln(1+(\frac{3}{2})^n))$

$\ln y =\lim_{n\rightarrow \infty}  \ln 2+ \frac{1}{n}\ln(1+(\frac{3}{2})^n))$

$\ln y = \ln 2+ \lim_{n\rightarrow \infty} \frac{1}{n}\ln(1+(\frac{3}{2})^n))$

$\text{On applying}$ $L’H\hat{o}pital’s$ $\text{rule,}$

$\ln y = \ln 2+ \lim_{n\rightarrow \infty} \frac{(\frac{3}{2})^n \ln \frac{3}{2}}{1+(\frac{3}{2})^n}$

$\ln y = \ln 2+ \lim_{n\rightarrow \infty} \frac{ \ln \frac{3}{2}}{1+\frac{1}{(\frac{3}{2})^n}}$

$\text{Since,}$ $\lim_{n\rightarrow \infty} (\frac{3}{2})^n \rightarrow \infty$

$\text{So,}$ $\ln y = \ln 2+ \ln \frac{3}{2}$

$\ln y = \ln (2* \frac{3}{2}) = \ln 3$

$y=3$

$\text{So, sequence $b_n$ converges to a value $3$}$