# Thread: Help Fast on Series/Sigma problem!!!

1. ## Help Fast on Series/Sigma problem!!!

the question is:

Show that if

(a(sub 0)/1) + (a(sub 1)/2) +... (a(sub n)/n+1) = 0

$\displaystyle a_0 /1 + a_1/2 +a_n/(n+1) = 0$

then

a(sub 0) + a(sub 1)*x +... a(sub n)*(x^n) = 0

$\displaystyle a_0 + x a_1 +.... x^n a_n = 0$

for some x in the interval [0,1]

I have not leaned anything about infinite series, so the solution doesnt pertain to that. I am guessing it has something to do with Sigma notation, and i can transfer the given equations to Sigma notation, but i have no idea where to go from there.

This is my guess at what it is saying in Sigma Notation, but I am not very good with it so I may be wrong

show that if

$\displaystyle \sum_{k=1}^{n}\,\,\frac{a_k}{k+1} =0$

then

$\displaystyle \sum_{k=1}^{n} a_k x^k =0$

for some x in [0,1]

I need help ASAP! thank you

2. Originally Posted by thunderhockey48
the question is:

Show that if

(a(sub 0)/1) + (a(sub 1)/2) +... (a(sub n)/n+1) = 0

$\displaystyle a_0 /1 + a_1/2 +a_n/(n+1) = 0$

then

a(sub 0) + a(sub 1)*x +... a(sub n)*(x^n) = 0

$\displaystyle a_0 + x a_1 +.... x^n a_n = 0$

for some x in the interval [0,1]

I have not leaned anything about infinite series, so the solution doesnt pertain to that. I am guessing it has something to do with Sigma notation, and i can transfer the given equations to Sigma notation, but i have no idea where to go from there.

This is my guess at what it is saying in Sigma Notation, but I am not very good with it so I may be wrong

show that if

$\displaystyle \sum_{k=1}^{n}\,\,\frac{a_k}{k+1} =0$

then

$\displaystyle \sum_{k=1}^{n} a_k x^k =0$

for some x in [0,1]

I need help ASAP! thank you
Let $\displaystyle f(x)=\sum_{k=1}^{n} a_k x^k$

Then $\displaystyle \int_0^1 f(x)dx =\sum_{k=1}^{n} a_k \int_0^1x^kdx=\sum_{k=1}^{n}\,\,\frac{a_k}{k+1} =0$

Hence, if a function has zero 'area' it must be zero for at least one value in that interval.

3. how did you get $\displaystyle \int_0^1 f(x)dx =\sum_{k=1}^{n} a_k \int_0^1x^kdx$ ? then to get $\displaystyle \sum_{k=1}^{n} a_k \int_0^1x^kdx=\sum_{k=1}^{n}\,\,\frac{a_k}{k+1}$ did you just integrate $\displaystyle \int_0^1x^kdx$ to get $\displaystyle x^{k+1} / k+1$, then you just multiplied $\displaystyle \sum_{k=1}^{n} a_k$, and since $\displaystyle \sum_{k=1}^{n}\,\,\frac{a_k}{k+1} =0$ then all of it =0? I can kinda follow how you got it, but not completely... thanks for the immediate reply!

4. Look at what I called f(x) and pass the integral through the FINITE sum
Next integrate AND plug in the value 0 and 1 for x.
This is a definite integral.

Ignore the summation...

Let $\displaystyle f(x)= a_1x+a_2x^2+\cdots +a_nx^n$

Then $\displaystyle \int_0^1 f(x)dx = a_1 \int_0^1xdx +a_2 \int_0^1x^2 dx +\cdots +a_n \int_0^1x^ndx$

Integrate each one and plug in the 0 and the 1.

You will get what I stated earlier tonight, that...

Then $\displaystyle \int_0^1 f(x)dx =\sum_{k=1}^{n}\,\,\frac{a_k}{k+1} =0$

Hence, if a continuous function has zero 'area' it must be zero for at least one value in that interval.
(This can be proved via contradiction. If a function doesn't have a zero, then it's strictly postive or strictly negative.
In those two case the 'area' will be postive or negative, BUT NOT zero.)

5. Oh, thank you so much! I understand it now. I didn't realize that you could pass the summation through the integral, but your proof below makes it clear. I was forgetting that it was a definite integral. You are a lifesaver!

6. Originally Posted by thunderhockey48
Oh, thank you so much! I understand it now. I didn't realize that you could pass the summation through the integral, but your proof below makes it clear. I was forgetting that it was a definite integral. You are a lifesaver!
You can switch the order since this is a finite summation.
IF it was an infinite summation then you need to worry, because you're switching the order of two limits.
One limit is your infinite summation, the other is the integral.
But here you have just one limit, your integral.

7. Did you get this problem out of Rudin's "Principles of Mathematical Analysis" (Chapter 5, Question 4)?

Also, it's not a big deal, but I think your summations should be going from 0 to n and not 1 to n.

8. I ignored the first stuff and used his/her TeX.