So for f(z)=1/(z+1)−1/(z+4), I am supposed to find the Maclaurin Series and give the region which it converges. Also, I am supposed to find f^(17)(0) without computing the derivatives. Find the region it converges and how to compute the 17th derivative.

Jazmin Clark

Jazmin Clark

Answered question

2022-08-11

So for f ( z ) = 1 z + 1 1 z + 4 , I am supposed to find the Maclaurin Series and give the region which it converges. Also, I am supposed to find f ( 17 ) ( 0 ) without computing the derivatives.
I know how to find the Maclaurin series, I am just struggling with finding the region it converges and how to compute the 17th derivative.

Answer & Explanation

kilinumad

kilinumad

Beginner2022-08-12Added 21 answers

Recall the maclaurin series of a function f is written n = 0 f ( n ) ( 0 ) n ! z n where f ( n ) is the nth derivative. So it's asking for the coefficient at n = 17. To find the radius of convergence, note that the geometric series is valid for | z | < 1.
muroscamsey

muroscamsey

Beginner2022-08-13Added 3 answers

Use the geometric series,
1 1 z = n = 1 z n .
Substitute z for z to get the series for 1 / ( 1 + z ). As for the other term, use the identity
1 4 + z = 1 / 4 1 + z / 4 .

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