Find the limits of the following sequences ({\cos((2n+1)\frac{\pi}{2})})_{n=1}^{\infty} ({\frac{\pi^n}{e^{2n+1}}})_{n=1}^{\infty} ({\frac{n^2+3}{n^3+n^2-1}})_{n=1}^{\infty} ({n\sin\frac{\pi}{n}})_{n=1}^{\infty}

Question

Find the limits of the following sequences

{({\cos ((2 n + 1) \frac{π}{2})})}_{n = 1}^{\infty}

{({\frac{π^{n}}{e^{2 n + 1}}})}_{n = 1}^{\infty}

{({\frac{n^{2} + 3}{n^{3} + n^{2} - 1}})}_{n = 1}^{\infty}

{({n \sin \frac{π}{n}})}_{n = 1}^{\infty}

Answer 1

1) ${({\cos ((2 n + 1) \frac{π}{2})})}_{n = 1}^{\infty}$
$lim_{n \to \infty} (\cos (2 n + 1) \frac{π}{2})$
$= \cos (\infty) \frac{π}{2}$
$= \cos \infty = O \neq$ $(∵ \cos \infty = n o t u n q i u e)$
$= lim_{n \to \infty} [\cos (2 n + 1) \frac{π}{2}] = o \neq$
2) ${({\frac{π^{n}}{e^{2 n + 1}}})}_{n = 1}^{\infty}$
$lim_{n \to \infty} \frac{π^{n}}{e^{2 n + 1}}$
$lim_{n \to \infty} \frac{π^{n}}{{(e^{2})}^{n} \times e}$
$lim_{n \to \infty} {(\frac{π^{n}}{e^{2}})}^{n} \times \frac{1}{e}$
$= {(\frac{π}{e^{2}})}^{\infty} \times \frac{1}{e} = 0$
$= lim_{n \to \infty} \frac{π^{n}}{e^{2 n + 1}} = 0$

Answer 2

(c)

{({\frac{n^{2} + 3}{n^{3} + n^{2} - 1}})}_{n = 1}^{\infty}

lim_{n \to \infty} n \sin \frac{π}{n}

\frac{\sin \frac{π}{n}}{\frac{1}{n}}

lim_{n \to \infty} (\frac{\sin \frac{π}{n}}{\frac{π}{n}})

= π \times 1

= π

(∵ lim_{x \to \infty} \frac{\sin \frac{1}{x}}{\frac{1}{x}}) = 1

= lim_{n \to \infty} (n \sin \frac{π}{n}) = π

(d)

{({n \sin \frac{π}{n}})}_{n = 1}^{\infty}

lim_{n \to \infty} \frac{n^{2} + 3}{n^{3} + n^{2} - 1}

lim_{n \to \infty} \frac{n^{2} (1 + \frac{3}{n^{2}})}{n^{3} (1 + \frac{1}{n} - \frac{1}{n^{3}})}

lim_{n \to \infty} \frac{\frac{1}{n} (1 + \frac{3}{n^{2}})}{1 + \frac{1}{n} - \frac{1}{n^{3}}}

\frac{\frac{1}{\infty} (1 + \frac{3}{\infty})}{1 + \frac{1}{\infty} - \frac{1}{\infty}}

= \frac{0 \times (1 + 0)}{1 + 0 - 0}

= 0

Find the limits of the following sequences ({\cos((2n+1)\frac{\pi}{2})})_{n=1}^{\infty} ({\frac{\pi^n}{e^{2n+1}}})_{n=1}^{\infty} ({\frac{n^2+3}{n^3+n^2-1}})_{n=1}^{\infty} ({n\sin\frac{\pi}{n}})_{n=1}^{\infty}

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