Find Maclaurin series for this f ( x ) = x 3 tan − 1...

garkochenvz

garkochenvz

2022-08-13

Find Maclaurin series for this
f ( x ) = x 3 tan 1 ( 2 x ) ; | x | < 1 2

Answer & Explanation

Kelsie Marks

Kelsie Marks

Expert

2022-08-14Added 17 answers

Hint: The Taylor series for n = 0 ( 1 ) n x 2 n + 1 2 n + 1 when | x | 1 .
Nica2t

Nica2t

Expert

2022-08-15Added 4 answers

For the geometric series with common ratio x 2 we have:
n = 0 ( x 2 ) n = 1 x 2 + x 4 x 6 + = 1 1 + x 2
for | x | < 1.. Integrate
C + n = 0 ( 1 ) n x 2 n + 1 2 n + 1 = tan 1 x .
Plug in x = 0 to discover that C = 0.. Replace x by 2 x to get the series for tan 1 2 x then multiply the whole lot by x 3 .

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