Homework Statement
I am trying to prove that the length of a helix can be represented by 2\pi=\sqrt{a^2+b^2}
Homework Equations
The Attempt at a Solution
I have the following so far:
If the helix can be represented by h(t)=a\cdot cos(t)+a\cdot sin(t)+b(t)
Then the length is...
What is a "Small" Number?
Homework Statement
I am given the equation, and asked to find an approximation (using Taylor's Formula):
E=\frac{q}{z^2(1-d/z)^2}-\frac{q}{z^2(1+d/z)^2}
I am also told that I can assume "z is much larger than d, so d/z is small."
Does this mean that I can assume d/z...
I've seen this "set x=1/2" technique used both in class and online. How are we just able to set this value of x? Does it derive from some theorem or something?
I'll definitely do that, but I'd just like to know the logic behind it, and why it works (or that there is a theorem that this comes...
Homework Statement
I feel bad asking another question after I just asked one yesterday, but I'm really close this time, I think!
I have:
\sum_{n=2}^{\infty}\frac{n^2-n}{2^n}
And need to find the sum.
Homework Equations
\sum_{n=1}^{\infty}nx^{n-1}=\frac{1}{(1-x)^2}
The Attempt at a...
I get that part, but what I don't get is how to convert the sum I get from that series back into what I have for the original.
So if I apply the sum formula to the derivative form (nx^{n-1}), using x as r and n as a in \frac{a}{1-r} and find this sum from the derivative, can I just derive the...
Awesome! But now I'm stuck on one more thing... I can't actually find the sum of the geometric series for which I have the derivative, because the summation starts at 2. Can I just find the sum as if it started at 0 and then subtract those first two terms?
Then I'll have that my original sum...
Oh! It's the derivative of nx^{n-1} which I can evaluate as a geometric series. How do you just "see" this stuff in these equations?
So if I integrate it, the definite integral will equal the sum of the geometric series generated by integration.
Does this mean that if I multiply the sum of...
I get 2x^2+6x^3+12x^4+...+n(n-1)x^n
Oh, so I did make a mistake there. But factoring out x2 still doesn't illuminate anything for me. By doing that, I now have n(n-1)x^2\cdot x^{n-2}.
I still see no way to rewrite this as a geometric series. Sorry if I'm being a bother; but basically the...
Yeah, this was option #2 in my list. I tried that, and found that only x was factorable, thus creating n(n-1)x\cdot x^{n-1}
And this is closer to the geometric form I was looking for, but I don't see how I can apply the geometric series sum formula to it in this form; I need to get rid of the...
Hmm.. I'm trying to figure out what you mean by expand.
Do you want me to:
- Expand the terms in the sum so that I get n^2-nx^n
- Expand the series for a few iterations (i.e. write it down for n=2,3,4, etc.)
or
- Perform a Taylor expansion on it where I use the formula...
Homework Statement
I am asked to find the sum of the series:
\sum_{n=2}^{\infty}n(n-1)x^n, |x|<1
Homework Equations
The Attempt at a Solution
We've learned a bit about Taylor series, so I could approximate this answer by taking and summing the first few derivatives. However, I suspect a more...
Okay, thanks guys :) That makes perfect sense to me if I accept the fact that convergence is guaranteed along (-4,4). However I don't think I quite understand why convergence is guaranteed along that interval.
Is it as simple as saying that since it's centered around 0 and converges at x=-4...
Homework Statement
I am asked to comment on the convergence/divergence of three series based on some given information about a power series:
\sum_{n=0}^{\infty}c_nx^n converges at x=-4 and diverges x=6.
I won't ask for help on all of the series, so here's the first one...