Evaluate the limit of the following sequence. a_n = cos (0.95^n) + 4^n + 9^n/36^

Question

Evaluate the limit of the following sequence. a_n = cos (0.95^n) + 4^n + 9^n/36^n Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The sequence is not monotonic. The least value by which it is bounded above is M = The limit of the sequence is (Type exact answers.) B. The sequence is monotonic. The least value by which it is bounded above is M = The limit of the sequence is (Type exact answers.) C. The sequence is monotonic. There is no number by which it is bounded above. The limit is (Type an exact answer.) D. The sequence is not monotonic. The limit does not exist.

Explanation / Answer

to fond limit of sequence let's conisder n = infinity

cos (infinity) somewhere between -1 to 1

(4n +9n)/36n when n = inifnity using L-hospital rule and differnetiating both side

we got = (4+9)/36 = 13/36

So least possible value -1 +13/36 =-0.6389

for n (-inifnity) = to find value of (4-n +9-n)/36-n

= {36n(4n +9n)}/4n9n = inifnty

So function is montonus decreasing with least value = -0.6389

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