Consider tan-1 x = C. +∞ n=0 (-1)¹x²n+1 2n + 1 for all x € [-1,1]. a. By differentiating tan−¹(x²), find a power series representation of f(x) € (-1,1). +∞o b. Use the result in item a. to find the exact value of n=0 (−1)n+1 16n Approximate (tan-¹¹) using a 4th degree Maclaurin polynomial. Hint: Using the power series representation for tan¹ given above, write a tan -1 2x 1+x¹ that is valid for all as a power series.
Consider tan-1 x = C. +∞ n=0 (-1)¹x²n+1 2n + 1 for all x € [-1,1]. a. By differentiating tan−¹(x²), find a power series representation of f(x) € (-1,1). +∞o b. Use the result in item a. to find the exact value of n=0 (−1)n+1 16n Approximate (tan-¹¹) using a 4th degree Maclaurin polynomial. Hint: Using the power series representation for tan¹ given above, write a tan -1 2x 1+x¹ that is valid for all as a power series.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider tan-1 x =
+∞
Σ
n=0
(-1)²n+1
2n + 1
for all x € [-1,1].
a. By differentiating tan-¹ (2²), find a power series representation of f(x)
€ (-1,1).
c. Approximate tan
b. Use the result in item a. to find the exact value of
+∞
n=0
(−1)n+1
16n
=
2x
1+xª
that is valid for all
¹) using a 4th degree Maclaurin polynomial.
Hint: Using the power series representation for tan given above, write x tan¹ as a power series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F498f8959-9b23-405c-ba0e-9300b89e04ce%2F9c8396f7-177b-4929-bf87-7ac520c5b478%2Faaed728_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider tan-1 x =
+∞
Σ
n=0
(-1)²n+1
2n + 1
for all x € [-1,1].
a. By differentiating tan-¹ (2²), find a power series representation of f(x)
€ (-1,1).
c. Approximate tan
b. Use the result in item a. to find the exact value of
+∞
n=0
(−1)n+1
16n
=
2x
1+xª
that is valid for all
¹) using a 4th degree Maclaurin polynomial.
Hint: Using the power series representation for tan given above, write x tan¹ as a power series.
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