Finding the value of combination using integration

[songoku]

Homework Statement

By using integration, find the value of
$$100C0+\frac{1}{2}100C1+\frac{1}{3}100C2+\frac{1}{4}100C3+...+\frac{1}{101}100C100$$

Homework Equations

Integration
Combination

The Attempt at a Solution

I don't even have any ideas to start. Can this really be solved by using integration?

Thanks

 

[haruspex]

The trick is to turn the sum into a power series, then perform a calculus operation on it which gets rid of those nasty 1/r factors.

 

[lurflurf]

recall that for k=1,2,...

$$\frac{1}{k}=\int_0^1 \! x^{k-1} \, \mathrm{dx} \\ \text{or} \\
\frac{1}{k}=\int_0^\infty \! e^{-k \, x} \, \mathrm{dx} $$

use binomial theorem

 

[songoku]

The trick is to turn the sum into a power series, then perform a calculus operation on it which gets rid of those nasty 1/r factors.

recall that for k=1,2,...

$$\frac{1}{k}=\int_0^1 \! x^{k-1} \, \mathrm{dx} \\ \text{or} \\
\frac{1}{k}=\int_0^\infty \! e^{-k \, x} \, \mathrm{dx} $$

use binomial theorem

I don't think I get both of your hints

The question can be written as:

$$\Sigma_{r=0}^{100}~100Cr . \frac{1}{r+1}$$

Then, by using Maclaurin series:
$$\frac{1}{r+1}=1-r+r^2-r^3+...=\Sigma_{i=0}^{∞}(-r)^i$$

I am not sure what I am doing and the direction I am heading to...

Thanks

 

[haruspex]

I don't think I get both of your hints

The question can be written as:

$$\Sigma_{r=0}^{100}~100Cr . \frac{1}{r+1}$$

My hint was to introduce a variable, something like (but not exactly):
$$\Sigma_{r=0}^{100}~100Cr . \frac{1}{r+1} s^r$$

 

[ehild]

Use binomial theorem to write up 100 (1+x)¹⁰⁰. Then integrate it, from 0 to 1. What do you get?


ehild

 

[lurflurf]

my hint was

$$\sum_{k=0}^{100} {100 \choose k} \, \frac{1}{k+1}=\sum_{k=0}^{100} {100 \choose k} \, \int_0^1 \! x^k \, \mathrm{dx}=\int_0^1\sum_{k=0}^{100} {100 \choose k} x^k \, 1^{100-k} \, \mathrm{dx}$$
use the binomial theorem

 

[songoku]

I am really sorry for taking a long time to reply

My hint was to introduce a variable, something like (but not exactly):
$$\Sigma_{r=0}^{100}~100Cr . \frac{1}{r+1} s^r$$

Sorry I still don't get the hint

my hint was

$$\sum_{k=0}^{100} {100 \choose k} \, \frac{1}{k+1}=\sum_{k=0}^{100} {100 \choose k} \, \int_0^1 \! x^k \, \mathrm{dx}=\int_0^1\sum_{k=0}^{100} {100 \choose k} x^k \, 1^{100-k} \, \mathrm{dx}$$

use the binomial theorem

I don't get the last part. Where 1^100-k comes from? And also how to use the binomial theorem for your equation? I only know that binomial theorem is used for expanding. In your equation, what is the term that can be expanded using binomial theorem?

Use binomial theorem to write up 100 (1+x)¹⁰⁰. Then integrate it, from 0 to 1. What do you get?


ehild

Wow, expanding (1 + x)¹⁰⁰ then integrating it from 0 to 1 gives me the same result as calculating

$$\sum_{k=0}^{100} {100 \choose k} \, \frac{1}{k+1}$$

So, I can write

$$\sum_{k=0}^{100} {100 \choose k} \, \frac{1}{k+1}=\int_{0}^{1}{(1+x)^{100}}dx$$

How can we know such sigma form can be written in simple integral form? Do I have to memorize certain form or expression?

Thanks

 

[Dick]

I am really sorry for taking a long time to reply



Sorry I still don't get the hint



I don't get the last part. Where 1^100-k comes from? And also how to use the binomial theorem for your equation? I only know that binomial theorem is used for expanding. In your equation, what is the term that can be expanded using binomial theorem?



Wow, expanding (1 + x)¹⁰⁰ then integrating it from 0 to 1 gives me the same result as calculating

$$\sum_{k=0}^{100} {100 \choose k} \, \frac{1}{k+1}$$

So, I can write

$$\sum_{k=0}^{100} {100 \choose k} \, \frac{1}{k+1}=\int_{0}^{1}{(1+x)^{100}}dx$$

How can we know such sigma form can be written in simple integral form? Do I have to memorize certain form or expression?

Thanks

Just use the binomial theorem on (1+x)^100 and integrate terms. Knowing the binomial theorem is all you have to memorize. Knowing how it relates to to your problem is more of a hunch thing.

 

[lurflurf]

I don't get the last part. Where 1^100-k comes from? And also how to use the binomial theorem for your equation? I only know that binomial theorem is used for expanding. In your equation, what is the term that can be expanded using binomial theorem?

I am using the binomial theorem backwards to combine the terms. I insert 1^(100-k) to make the equation exactly fit the binomial theorem. Of course 1^(100-k)=1 so it does not change anything.
$$\sum_{k=0}^{100} {100 \choose k} x^k \, 1^{100-k}=(1+x)^{100}$$

 

[songoku]

Just use the binomial theorem on (1+x)^100 and integrate terms. Knowing the binomial theorem is all you have to memorize. Knowing how it relates to to your problem is more of a hunch thing.

I am using the binomial theorem backwards to combine the terms. I insert 1^(100-k) to make the equation exactly fit the binomial theorem. Of course 1^(100-k)=1 so it does not change anything.
$$\sum_{k=0}^{100} {100 \choose k} x^k \, 1^{100-k}=(1+x)^{100}$$

Oh ok. Now I can see clearer the connection between stating the question in sigma form and then introducing integration to solve the problem.

Thanks a lot for all the help I got in this thread