1. ## Binomial coefficients

Prove that for all positive integers n > 1,

$\displaystyle \sum_{k=0}^n\frac{1}{k+1}\dbinom{n}{k}(-1)^{k+1}=0$

My attempt:

$\displaystyle (1+x)^n=\sum_{k=0}^n\dbinom{n}{k}x^k$

Integrating both sides,

$\displaystyle \frac{(1+x)^{n+1}}{n+1}=\sum_{k=0}^n\dbinom{n}{k} \frac{x^{k+1}}{k+1}+C$

Substituting x = 0,

$\displaystyle C=\frac{1}{n+1}$

$\displaystyle \frac{(1+x)^{n+1}}{n+1}=\sum_{k=0}^n\dbinom{n}{k} \frac{x^{k+1}}{k+1}+\frac{1}{n+1}$

Substituting x = -1,

$\displaystyle 0=\sum_{k=0}^n\dbinom{n}{k} \frac{(-1)^{k+1}}{k+1}+\frac{1}{n+1}$

$\displaystyle \sum_{k=0}^n\frac{1}{k+1}\dbinom{n}{k}(-1)^{k+1}=-\frac{1}{n+1}$

What am I doing wrong?

2. ## Re: Binomial coefficients

On second thoughts, I'm beginning to think that the book may be wrong and I may be right.

3. ## Re: Binomial coefficients

You are right and the question is wrong. You can check this by testing the formula with n=2. In that case, the sum becomes

$\displaystyle \sum_{k=0}^2\frac{1}{k+1}\dbinom{2}{k}(-1)^{k+1} = 1(-1) + \tfrac22(1) +\tfrac13(-1) = -\tfrac13.$

That agrees with your answer $\displaystyle -\tfrac{1}{n+1}$ (with n=2), but not with the given answer 0.

My guess is that the person who set this question forgot the constant when doing the integration. Congratulations to you for remembering that this is necessary.

4. ## Re: Binomial coefficients

Originally Posted by chisigma
The correct procedure should be...

$\displaystyle (1+t)^{n} = \sum_{k=0}^{n} \binom{n}{k}\ t^{k}$ (1)

... then integrating both terms from 0 to x...

$\displaystyle \frac{(1+x)^{n+1}}{n+1} = \sum_{k=0}^{n} \binom{n}{k}\ \frac{x^{k+1}}{k+1}$ (2)

... then setting $\displaystyle x=-1$...

$\displaystyle \sum_{k=0}^{n} \binom{n}{k}\ \frac{(-1)^{k+1}}{k+1} =0$ (3)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
After you integrated from 0 to x, you did not substitute for the lower limit 0.

5. ## Re: Binomial coefficients

Originally Posted by alexmahone
After you integrated from 0 to x, you did not substitute for the lower limit 0.
I discovered that and I canceled my post before reading Your post...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$