Find a formula for ¹ - 1²/ ( 7 ) + 3² ( 2 ) - ··· + + (−1)n. 2+1(1).
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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![**Question:**
Find a formula for
\[ 1 - \frac{1}{2} \binom{n}{1} + \frac{1}{3} \binom{n}{2} - \cdots + (-1)^n \frac{1}{n + 1} \binom{n}{n}. \]
**Explanation:**
The provided mathematical expression alternates in sign and includes binomial coefficients. The general term in the sequence is given by:
\[ (-1)^k \frac{1}{k + 1} \binom{n}{k}, \]
where \( k \) ranges from 0 to \( n \). The first term of the sequence is 1, the next term is subtracted, and so on, following the alternating pattern \((-1)^k\).
Each binomial coefficient \( \binom{n}{k} \) stands for the number of ways to choose \( k \) elements from \( n \) elements and can be calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n - k)!}. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd04de2b6-c3ee-4b26-bb77-15432c184cae%2F671be920-b4cf-45e7-abdd-fef3f972b9a7%2Fmg81uws_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
Find a formula for
\[ 1 - \frac{1}{2} \binom{n}{1} + \frac{1}{3} \binom{n}{2} - \cdots + (-1)^n \frac{1}{n + 1} \binom{n}{n}. \]
**Explanation:**
The provided mathematical expression alternates in sign and includes binomial coefficients. The general term in the sequence is given by:
\[ (-1)^k \frac{1}{k + 1} \binom{n}{k}, \]
where \( k \) ranges from 0 to \( n \). The first term of the sequence is 1, the next term is subtracted, and so on, following the alternating pattern \((-1)^k\).
Each binomial coefficient \( \binom{n}{k} \) stands for the number of ways to choose \( k \) elements from \( n \) elements and can be calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n - k)!}. \]
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