Write the power series for (1 + x)* in terms of binomial coefficients. 00 (1 + x) .
Write the power series for (1 + x)* in terms of binomial coefficients. 00 (1 + x) .
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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![Write the power series for \( (1 + x)^k \) in terms of binomial coefficients.
\[
(1 + x)^k = \sum_{n = 0}^{\infty} \binom{k}{n} x^n
\]
This equation represents the binomial theorem, where \((1 + x)^k\) is expressed as an infinite series. The binomial coefficient \(\binom{k}{n}\) is used in each term of the series, indicating the combination of \(k\) items taken \(n\) at a time, showing how the terms of this expansion are structured.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15d21cd-3875-4270-92b4-6749e1bc7456%2F02cc6a49-3399-4714-a95d-b0c14196b8ae%2F3deuyqc_processed.png&w=3840&q=75)
Transcribed Image Text:Write the power series for \( (1 + x)^k \) in terms of binomial coefficients.
\[
(1 + x)^k = \sum_{n = 0}^{\infty} \binom{k}{n} x^n
\]
This equation represents the binomial theorem, where \((1 + x)^k\) is expressed as an infinite series. The binomial coefficient \(\binom{k}{n}\) is used in each term of the series, indicating the combination of \(k\) items taken \(n\) at a time, showing how the terms of this expansion are structured.
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