
Thomas Z.
asked 11/19/20Find the Maclaurin series
Find in the Maclaurin series ∑ n=0 ∞ an for ƒ(x) = 2x6 cos (x9) and find the interval on which the expansion is valid.
1 Expert Answer
As the Maclaurin series for cos(x) is ∑n=0∞(-1)n x2n/2n!, the Maclaurin series for cos(x9) is
∑n=0∞(-1)n (x9)2n/2n! = ∑n=0∞(-1)n x18n/2n!
Thus, the Maclaurin series for 2x6cos(x9) is
∑n=0∞2x6(-1)n x18n/2n! =
∑n=0∞2(-1)n x18n+6/2n!
Then, the ratio of successive terms is -x18/((2n+2)(2n+1))
The ratio goes to 0 as n goes to infinity for all x
The interval of convergence is (-∞, ∞)
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Stanton D.
Hi Thomas Z., There's a disconnect in how you stated your problem, you shifted from n to x without stating how. But it looks as if you want |x|<1 , that will quench your series. An integral expression might be nicer? (But the polynomial covers the discrete sum exactly). -- Cheers, -- Mr. d.11/20/20