Chapter 8: Q. 73 (page 702)

Prove that if the series ${鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$ and ${鈭�}_{k = 0}^{鈭�} b_{k} x^{k}$ both converge to the same sum for every value of x in some nontrivial interval, then a_k = b_k for every nonnegative integer k.

Short Answer

Expert verified

$The series {鈭�}_{k = 0}^{鈭�} (a_{k} - b_{k}) x^{k} is a maclaurin series for the zero function . Thus, each coefficient a_{k} - b_{k} = 0 and hence, a_{k} = b_{k} .$

Step by step solution

Step 1. Given information is:

${鈭�}_{k = 0}^{鈭�} a_{k} x^{k} and {鈭�}_{k = 0}^{鈭�} b_{k} x^{k} both converge to the same sum for every value of x in some nontrivial interval .$

Step 2. Proving ak = bk

$The two series are the Maclaurin series for some function f (x) . Thus, the series {鈭�}_{k = 0}^{鈭�} (a_{k} - b_{k}) x^{k} is a maclaurin series for the zero function . Thus, each coefficient a_{k} - b_{k} = 0 . Therefore, we can say that a_{k} = b_{k} for every value of k .$

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91影视

Prove that if the series ${鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$ and ${鈭�}_{k = 0}^{鈭�} b_{k} x^{k}$ both converge to the same sum for every value of x in some nontrivial interval, then a_k = b_k for every nonnegative integer k.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Proving ak = bk

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91影视

Prove that if the series 鈭�k=0鈭�akxkand 鈭�k=0鈭�bkxkboth converge to the same sum for every value of x in some nontrivial interval, then ak = bk for every nonnegative integer k.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Proving ak = bk

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Probability and Statistics

Geometry

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Prove that if the series ${鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$ and ${鈭�}_{k = 0}^{鈭�} b_{k} x^{k}$ both converge to the same sum for every value of x in some nontrivial interval, then a_k = b_k for every nonnegative integer k.