Q. 3 - Calculus

Question

If a sequence $a_1,a_2,a_3,...,a_k,...$approaches a real-number limit as $k\to \infty$, then the sequence $\left\{a_k\right\}$ converges. If the terms of the sequence do not get arbitrarily close to some real number, then the sequence diverges. Determine the general form $\left\{a_k\right\}$ for each of the following sequences, and then use L’Hospital’s rule to determine whether that sequence converges or diverges.

$\frac{1}{10},\frac{3}{100},\frac{7}{1000},\frac{15}{10,000},\frac{31}{100,000},\frac{63}{1,000,000},...$ 

Step-by-Step Solution

Answer

The given sequence converges.

Step 1. Given information.

Consider the given question,

$\frac{1}{10},\frac{3}{100},\frac{7}{1000},\frac{15}{10,000},\frac{31}{100,000},\frac{63}{1,000,000},...$

Step 2. Calculate the sequence.

A sequence say $\left\{a_n\right\}$is said to be convergent if the nth term of the sequence approaches unique finite as n approaches $\infty$

otherwise sequence said to be divergent.

From the given sequence, nth term is $\left\{a_n\right\}=\frac{\ln n}{\ln n+1} ...... \left(i\right)$.

On evaluating limit of equation (i) as $n\to \infty$,

$$\begin{gathered} \underset{n\to \infty }{\lim }\left\{a_n\right\}=\frac{2^n-1}{{10}^n} \\ =\underset{n\to \infty }{\lim }\frac{2^n\left(\ln 2\right)}{{10}^n\left(\ln 10\right)} \\ =\left(\frac{\ln 2}{\ln 10}\right)\underset{n\to \infty }{\lim }{\left(\frac{2}{10}\right)}^n \\ =\left(\frac{\ln 2}{\ln 10}\right)\left(0\right) \\ =0 \end{gathered}$$

Thus, $\underset{n\to \infty }{\lim }\left\{a_n\right\}=0$.

Therefore, the given sequence is convergent.