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Prove n (1+x)n-1= Σ c(n, k) k xk-1. k=1

Prove n (1+x)n-1= Σ c(n, k) k xk-1. k=1

Advanced Engineering Mathematics
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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**Prove** \( n (1 + x)^{n-1} = \sum_{k=1}^{n} c(n, k) \, k \, x^{k-1} \). This equation involves demonstrating the equality between the expression on the left hand side and the summation on the right. The left hand side includes the term \( n (1 + x)^{n-1} \), which represents an algebraic expression involving a binomial expansion. The right hand side consists of a summation from \( k = 1 \) to \( n \), with each term being a product of a coefficient \( c(n, k) \), the variable \( k \), and \( x^{k-1} \).
Transcribed Image Text:**Prove** \( n (1 + x)^{n-1} = \sum_{k=1}^{n} c(n, k) \, k \, x^{k-1} \). This equation involves demonstrating the equality between the expression on the left hand side and the summation on the right. The left hand side includes the term \( n (1 + x)^{n-1} \), which represents an algebraic expression involving a binomial expansion. The right hand side consists of a summation from \( k = 1 \) to \( n \), with each term being a product of a coefficient \( c(n, k) \), the variable \( k \), and \( x^{k-1} \).
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