does the sequence {an} converge or diverge? find the limit if the sequence is convergent. an = 5n - √(25n²…
does the sequence {an} converge or diverge? find the limit if the sequence is convergent. an = 5n - √(25n² - 5n) select the correct choice below and, if necessary, fill in the answer box to complete the choice. a. the sequence converges to lim an = (type an integer or a simplified fraction.) b. the sequence diverges.
Answer
Explanation:
Step1: Rationalize the expression
Multiply and divide by the conjugate $5n+\sqrt{25n^{2}-5n}$: [ \begin{align*} a_{n}&=5n - \sqrt{25n^{2}-5n}\ &=\frac{(5n - \sqrt{25n^{2}-5n})(5n+\sqrt{25n^{2}-5n})}{5n+\sqrt{25n^{2}-5n}}\ &=\frac{(5n)^{2}-( \sqrt{25n^{2}-5n})^{2}}{5n+\sqrt{25n^{2}-5n}}\ &=\frac{25n^{2}-(25n^{2}-5n)}{5n+\sqrt{25n^{2}-5n}}\ &=\frac{5n}{5n+\sqrt{25n^{2}-5n}} \end{align*} ]
Step2: Divide numerator and denominator by $n$
[ \begin{align*} \lim_{n\rightarrow\infty}a_{n}&=\lim_{n\rightarrow\infty}\frac{5n/n}{(5n+\sqrt{25n^{2}-5n})/n}\ &=\lim_{n\rightarrow\infty}\frac{5}{5+\sqrt{25-\frac{5}{n}}} \end{align*} ]
Step3: Evaluate the limit
As $n\rightarrow\infty$, $\frac{5}{n}\rightarrow0$. [ \begin{align*} \lim_{n\rightarrow\infty}\frac{5}{5+\sqrt{25-\frac{5}{n}}}&=\frac{5}{5 + \sqrt{25-0}}\ &=\frac{5}{5 + 5}\ &=\frac{5}{10}\ &=\frac{1}{2} \end{align*} ]
Answer:
A. The sequence converges to $\lim_{n\rightarrow\infty}a_{n}=\frac{1}{2}$