a) We use the definition of speed:
v = delta_L/delta_t
delta_t = delta_L/v = 45000 m/(0.99540*3*10^8 m/s) = 1.55*10^-4 s
b) We use the length contraction equation:
delta_L = L_0*sqrt(1-v^2/c^2)
L_0 = delta_L/sqrt(1-v^2/c^2) = 45000 m/sqrt(1-0.99540^2) = 469698 m
However, the solution shows...
I solved the integral by two different methods and I get different answers.
Method 1:
∫dx/(1-x) = -∫-dx/(1-x), u=1-x, du=-dx
∫dx/(1-x) = -∫du/u = -ln|u| = -ln|1-x|
Method 2:
∫-dx/(x-1) = -∫dx/(x-1), u=x-1, du=dx
∫-dx/(x-1) = -∫du/u = -ln|u| = -ln|x-1|
What am I doing wrong?
We transform the series into a power series by a change of variable:
y = √(x2+1)
We have the following after substituting:
∑(2nyn/(3n+n3))
We use the ratio test:
ρn = |(2n+1yn+1/(3n+1+(n+1)3)/(2nyn/(3n+n3)| = |(3n+n3)2y/(3n+1+(n+1)3)|
ρ = |(3∞+∞3)2y/(3∞+1+(∞+1)3)| = |2y|
|2y| < 1
|y| = 1/2...
∑((√(x2+1))n22/(3n+n3))
We use the ratio test:
ρn = |2(3n+n3)√(x2+1)/(3n+1+(n+1)3)|
ρ = |2√(x2+1)|
ρ < 1
|2√(x2+1)| < 1
No "x" satisfies this expression, so I conclude the series doesn't converge for any "x". However the answer in the book says the series converges for |x| < √(5)/2. What am...
∑(x2n/(2nn2))
We use the ratio test:
ρn = |(x2n2/(2(n+1)2)|
ρ = |x2/2|
ρ < 1
|x2| < 2
|x| = √(2)
We investigate the endpoints:
x = 2:
∑(4n/(2nn2) = ∑(2n/n2))
We use the preliminary test:
limn→∞ 2n/n2 = ∞
Since the numerator is greater than the denominator. Therefore, x = 2 shouldn't be...
I found that ρn = √(2n+1)/(n+1).
Then, I found ρ = lim when n→∞ |(1/n) (√(2n+1))/((1/n) (n+1))| = 0
Based on this result I concluded the series converges; however, the book answer says it diverges. What am I doing wrong?
After evaluating the integral I found the following:
(1/3)tan-1(e∞/3) = (1/3)tan-1(∞) = (1/3)(nπ/2), where n is an odd number. In this case I found multiple solutions to the problem. How do you prove it converges?
I got the following expression:
-(1/4)ln((n+2)/(n-2))
When I substitute "∞" in the expression I found it undefined. However, the book says the series converges. What am I doing wrong?
If n is ∞, then ln (n) = ln (∞) = ∞
Then, 1/∞ = 0
Any number raised to "0" = 1, so the answer should be 1. However the book says the answer is e2. Could you provide me some help?
Problem Statement: Derive the optical law of reflection. Hint: Let light go from the point A (x1, y1) to B (x2, y,2) via an arbitrary point P = (x, 0) on a mirror along the x axis. Set dt/dx = (n/c) dD/dx = 0, where D = distance APB, and show that then theta = phi.
Relevant Equations: t = nD/c...
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