Chapter 8: Q. 73 (page 702)

Prove that if the series $\sum_{k = 0}^{\infty} a_{k} x^{k}$ and $\sum_{k = 0}^{\infty} b_{k} x^{k}$ 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

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

Step by step solution

Step 1. Given information is:

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

Step 2. Proving ak = bk

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

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Prove that if the series $\sum_{k = 0}^{\infty} a_{k} x^{k}$ and $\sum_{k = 0}^{\infty} b_{k} x^{k}$ 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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Prove that if the series ∑k=0∞akxkand ∑k=0∞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

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Prove that if the series $\sum_{k = 0}^{\infty} a_{k} x^{k}$ and $\sum_{k = 0}^{\infty} b_{k} x^{k}$ 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.