Can you solve number 5, 6, 7 with sequence test ? what is missing from number 9

Question

Can you solve number 5, 6, 7 with sequence test ? what is missing from number 9 ?

uences 365 5. Determine the limit ol the sequence y or show that the sequence diverges Gustifying each step using the appropriate Limit Laws or Theorems). or show that the sequence diverges 6. Determine the limit of the sequence a A liustifying each step using the appropriate Limit Laws or Theorems). S an 7. Determine the limit of the sequence b en -n or S that the sequence diverges how Gustifying each step using the appropriate Limit Laws or Theorems). an 8. Determine the limit 3- An or show that the sequence diverges of the sequence b bin 10 di Venues M 3n' s strictly increasing. Find an upper bound. 9. Show that a n? 2 I M

Explanation / Answer

5) we have given yn=en/2n

let an =en/2n and an+1=en+1/2n+1

Applying series ratio test

summation of (n=0 to infinity)en/2n

L=lim n-->infinity|an+1/an|

=lim n-->infinity|(en+1/2n+1)*(2n/en)|

=lim n-->infinity|(e/2)

L=e/2

L>1

so given series is diverges

6) we have given an=sqrt(n)/(sqrt(n)+4)

summation of (n=0 to infinity) sqrt(n)/(sqrt(n)+4)

by series divergence test if lim n-->infinity an not equal to 0, then the series is divergence.

lim n-->infinity (sqrt(n)/(sqrt(n)+4)

=lim n-->infinity (sqrt(n)/sqrt(n)(1+4/sqrt(n)))

=lim n-->infinity (1/(1+4/sqrt(n)))

lim n-->infinity (sqrt(n)/(sqrt(n)+4)=1

so given series divergence

7) we have given bn=en^2-n

let bn=en^2-n and bn+1=e(n+1)^2-(n+1)

Applying series ratio test

summation of (n=0 to infinity) en^2-n

L=lim n-->infinity |bn+1/ bn|=lim n-->infinity |e(n+1)^2-(n+1)/ en^2-n |

=lim n-->infinity |e(n^2+1+2n-n-1-n^2+n| since x^a/x^b=x^(a-b)

=lim n-->infinity |e2n |

=infinity

L>1

so given series diverges

9) we have given an=3n2/(n2+2)

if strictly increasing an<=an+1

put n=1

a1=3/3=1, a2=2,a3=27/11 and so on

the given series is strictly increasing

lim n-->infinity 3n2/(n2+2)=lim n-->infinity 3n2/n2(1+2/n2)=lim n-->infinity 3/(1+2/n2) =3

The given series upper bound is 3

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