Exercise 6.6.10. Consider f(x) = 1/√1-x. (a) Generate the Taylor series for f centered at zero, and use Lagrange's Remainder Theorem to show the series converges to f on [0, 1/2]. (The case x < 1/2 is more straightforward while x = 1/2 requires some extra care.) What happens when we attempt this with x > 1/2? (b) Use Cauchy's Remainder Theorem proved in Exercise 6.6.9 to show the series representation for f holds on [0, 1).
Exercise 6.6.10. Consider f(x) = 1/√1-x. (a) Generate the Taylor series for f centered at zero, and use Lagrange's Remainder Theorem to show the series converges to f on [0, 1/2]. (The case x < 1/2 is more straightforward while x = 1/2 requires some extra care.) What happens when we attempt this with x > 1/2? (b) Use Cauchy's Remainder Theorem proved in Exercise 6.6.9 to show the series representation for f holds on [0, 1).
Chapter6: Exponential And Logarithmic Functions
Section6.2: Graphs Of Exponential Functions
Problem 52SE: Prove the conjecture made in the previous exercise.
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Show x<1/2, then x=1/2, and then x>1/2.
![Exercise 6.6.10. Consider f(x) = 1/√1-x.
(a) Generate the Taylor series for f centered at zero, and use Lagrange's
Remainder Theorem to show the series converges to f on [0, 1/2]. (The
case x < 1/2 is more straightforward while x = 1/2 requires some extra
care.) What happens when we attempt this with x > 1/2?
(b) Use Cauchy's Remainder Theorem proved in Exercise 6.6.9 to show the
series representation for f holds on [0, 1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6389447-1237-4af0-b5c6-eb1260425b55%2F2fa88489-a784-4c66-9f2c-7e88f5278723%2Fv2w4lrm_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 6.6.10. Consider f(x) = 1/√1-x.
(a) Generate the Taylor series for f centered at zero, and use Lagrange's
Remainder Theorem to show the series converges to f on [0, 1/2]. (The
case x < 1/2 is more straightforward while x = 1/2 requires some extra
care.) What happens when we attempt this with x > 1/2?
(b) Use Cauchy's Remainder Theorem proved in Exercise 6.6.9 to show the
series representation for f holds on [0, 1).
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