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}

Ethen Wong 2022-01-24 Answered
Find the limits of the following sequences
({cos((2n+1)π2)})n=1
({πne2n+1})n=1
({n2+3n3+n21})n=1
({nsinπn})n=1
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Expert Answer

waijazar1
Answered 2022-01-25 Author has 13 answers

1)({cos((2n+1)π2)})n=1
limn(cos(2n+1)π2)
=cos()π2
=cos=O (cos=not unqiue)
=limn[cos(2n+1)π2]=o
2)({πne2n+1})n=1
limnπne2n+1
limnπn(e2)n×e
limn(πne2)n×1e
=(πe2)×1e=0
=limnπne2n+1=0

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bemolizisqt
Answered 2022-01-26 Author has 16 answers
(c)({n2+3n3+n21})n=1
limnnsinπn
sinπn1n
limn(sinπnπn)
=π×1
=π
(limxsin1x1x)=1
=limn(nsinπn)=π
(d)({nsinπn})n=1
limnn2+3n3+n21
limnn2(1+3n2)n3(1+1n1n3)
limn1n(1+3n2)1+1n1n3
1(1+3)1+11
=0×(1+0)1+00
=0
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