Calculate the summations

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Calculate the summations

Re: Calculate the summations

How far have you been able to work these? And are these 4 questions?

-Dan

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Re: Calculate the summations

the firs problem

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the second problem

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the third problem

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the last problem

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till the farther far

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i would also intressted on how to solve this :D

Re: Calculate the summations

Quote:

Originally Posted by WORLD

the firs problem

$\Sigma _{k = 0} ^{100} \frac{1}{k + 1} \left ( _k ^{100} \right )$

What does $\left ( _k ^{100} \right )$ mean?

-Dan

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Re: Calculate the summations

It’s mean

Re: Calculate the summations

could any one help me

Re: Calculate the summations

Quote:

Originally Posted by WORLD

the firs problem
Find $\sum_{k=0}^{100}\frac{1}{k+1} \binom{100}{k}^2$

Hello, WORLD!

This is a tough problem for the algebra forum. I'm going to use a little calculus. I hope you will still find my remarks useful.

Start with the Binomial Theorem:
$(1+x)^n = \sum_{i=0}^n \binom{n}{i} x^i$

Integrate both sides from 0 to x (this is the calculus part). The result is
$\frac{1}{n+1}[(1+x)^{n+1}-1] = \sum_{i=0}^n \frac{1}{i+1} \binom{n}{i} x^{i+1}$

Now multiply these two equations together. I'm going to replace i with j in the first equation, so it will be easier to keep everything straight.
(*).... $(1+x)^n \cdot \frac{1}{n+1}[(1+x)^{n+1}-1] = \sum_{j=0}^n \binom{n}{j} x^j \cdot \sum_{i=0}^n \frac{1}{i+1} \binom{n}{i} x^{i+1}$

Now consider the coefficient of $x^{n+1}$ in (*). The left-hand side of (*) is equal to
$\frac{1}{n+1}[(1+x)^{2n+1} - (1+x)^{n}] = \frac{1}{n+1} \left[ \sum_{i=0}^{2n+1} \binom{2n+1}{i} x^i - \sum_{i=0}^n \binom{n}{i}x^i \right]$
so the coefficient of $x^{n+1}$ is
$\frac{1}{n+1} \binom{2n+1}{n+1}$

On the right-hand side of (*), the coefficient of $x^{n+1}$ occurs when j = n-i, so the coefficient is
$\sum_{i=0}^n \binom{n}{n-i} \cdot \frac{1}{i+1} \binom{n}{i} = \sum_{i=0}^n \frac{1}{i+1} \binom{n}{i}^2$
since $\binom{n}{i} = \binom{n}{n-i}$ .

The coefficient of $x^{n+1}$ must be the same on both sides of the equation, so we have
$\frac{1}{n+1} \binom{2n+1}{n+1} = \sum_{i=0}^n \frac{1}{i+1} \binom{n}{i}^2$

Now set n = 100, and you're done.