Science:Math Exam Resources/Courses/MATH103/April 2014/Question 04 (b)

MATH103 April 2014

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[hide] Question 04 (b)
Let $(a_{n})_{n\geq 1}$ be a sequence, where $a_{n}=1+{\frac {n}{(n+1)!}}$ . Find the limit of the sequence if it converges, or show that it diverges, otherwise. Work must be shown for full marks. Simplify fully.

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[show] Hint 1
Expand the factorial and cancel a common term in the numerator and denominator.

[show] Hint 2
Keep in mind $\lim _{n\to \infty }1+b$ is equivalent to $1+\lim _{n\to \infty }b$

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[show] Solution
We have $\lim _{n\to \infty }a_{n}=1+\lim _{n\to \infty }{\frac {n}{(n+1)n(n-1)\ldots 2\cdot 1}}.$ Cancelling the common factor $n$ in the numerator and denominator, we get ${\frac {n}{(n+1)n(n-1)\ldots 2\cdot 1}}={\frac {1}{(n+1)(n-1)\ldots 2\cdot 1}}.$ As $n\to \infty$ , the numerator ( $1$ ) of the last expression is constant, while the denominator ( $(n+1)(n-1)\ldots 2\cdot 1$ ) tends to $\infty$ . Thus, $\lim _{n\to \infty }{\frac {n}{(n+1)n(n-1)\ldots 2\cdot 1}}=\lim _{n\to \infty }{\frac {1}{(n+1)(n-1)\ldots 2\cdot 1}}=0.$ We conclude that $\lim _{n\to \infty }a_{n}=1+0=1$ .