the question is:
Show that if
(a(sub 0)/1) + (a(sub 1)/2) +... (a(sub n)/n+1) = 0
then
a(sub 0) + a(sub 1)*x +... a(sub n)*(x^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
then
for some x in [0,1]
I need help ASAP! thank you
how did you get ? then to get did you just integrate to get , then you just multiplied , and since then all of it =0? I can kinda follow how you got it, but not completely... thanks for the immediate reply!
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
Then
Integrate each one and plug in the 0 and the 1.
You will get what I stated earlier tonight, that...
Then
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.)
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.