Wikipedia:Reference desk/Archives/Mathematics/2007 April 10
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April 10
editOperations on PowerSeries
editCan someone help explain how to do this.
Find the power series representaion for f(x) and specify the radius of convergence.
Thanks in advance! Azninvazn88 03:20, 10 April 2007 (UTC)
Maybe even how to do power series in general. Azninvazn88 03:33, 10 April 2007 (UTC)
- I assume you want a Taylor series? Around what point? x=0? Do you know the Taylor series for around x=0? If so, it's not too hard to manipulate it into the other one. --Spoon! 05:23, 10 April 2007 (UTC)
- Thanks for you guys' help! I understand how to do it now. Azninvazn88 19:07, 10 April 2007 (UTC)
Logrithm? Solve for x in this?
editI am tring to find formula to solve for x: (1+x)^6=4.5
This is to work out the eqivalent interest rate. Like the return on investment is 450% after 6 days. So the daily interest rate would be (1+x)^6 = 450%, but I don't know what to do.
This isn't homework its for a video game called Harvest Moon. I am trying to compare investments, like one investment is 450% in 6 days. Another is 1,000% in 10 days. I need the average daily rate to compare them. Like I know I could compare them using a general multiple, 6*10= 60 days. So 4.5^10=3,405,063 compared 10^6 = 1,000,000 , but I want a general per day rate so I can make comparisons between many many more investments, such as 4000% in 30 days, or 2200% in 16 days. etc.
Thanks for any help!--Dacium 05:02, 10 April 2007 (UTC)
- First take the 6th root of both sides:
(1+x)^6 = 4.5
1 + x = (4.5)^(1/6)
- Then subtract 1 from each side:
x = (4.5)^(1/6) - 1
- Post your answer and we'll check it for you. StuRat 05:12, 10 April 2007 (UTC)
- Thanks! I don't know what I was thinking! I got 28.5% for 450% in 6 days and 25.9% for 100% in 10 days. The answers for 60 days come to about 3,405,000 and 1,000,000 yay!. Yeah I had a mental block I forget that I could multiple the powers 6 * 1/6 to get to power 1.... and 1/6 duh. I feel stupid now :-)--Dacium 05:29, 10 April 2007 (UTC)
- Assuming you meant to say 1000% instead of 100%, your figures all check out. Good job ! StuRat 06:04, 10 April 2007 (UTC)
There are 6 solutions.
- (1) x = -2.28
- (2) x = -1.64 - 1.11i
- (3) x = -1.64 + 1.11i
- (4) x = -0.35 - 1.11i
- (5) x = -0.35 + 1.11i
- (6) x = 0.28
The really amazing thing is that a negative interest rate of -2.28 ( -228% ) can generate a positive return in 6 days!!! Cool!
- 202.168.50.40 23:37, 10 April 2007 (UTC)
- You can get a positive return on a negative interest rate in any even number of days, so long as it's not -100% interest, which will give you a return of 0. Here is an example: suppose you invest $1.00 at -200% interest per day. After one day you have $1.00 * -200% + $1.00 = -$1.00. After another day you have $-1.00 * -200% + $-1.00 = $1.00. Also, if the interest is between -100% and 0% the return will be positive, but below what you put in. For example: suppose you invest $1.00 at -50% interest per day. After one day you have $1.00 * -50% + $1.00 = $0.50.— Daniel 01:50, 11 April 2007 (UTC)
- Oh! Please don't stop there! You got my panties in a knot. Tell me what kind of returns I can expect with imaginary interest rate, both positive imaginary and negative imaginary interest rates. 202.168.50.40 02:55, 11 April 2007 (UTC)
- Imaginary interest rates — or exponential eigenvalues in general — correspond to oscillations: cycles in the complex plane. Considering only the real part will yield sine and cosine waves. Complex exponents, being a combination of such oscillations with normal exponential growth or decay, yield growing or shrinking waves; a negative real part (like this -2.28) should yield a decay, but we are effectively using Euler's method here and the method is unstable for the chosen time step. --Tardis 22:03, 11 April 2007 (UTC)
- A small correction to Daniel's comment: The interest rate must be less than -100%. Anything between -100% and 0% will give exponential decay. -- Meni Rosenfeld (talk) 16:33, 11 April 2007 (UTC)