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euclid3.14
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Hi, if an = (-1)^n-1 /n what is the infinite product of (1+an)
does working out (1+(a2n-1))(1+a2n) help? :uhh:
does working out (1+(a2n-1))(1+a2n) help? :uhh:
euclid3.14 said:does working out (1+(a2n-1))(1+a2n) help? :uhh:
euclid3.14 said:Thanks for the responce!
I got it to eqaul 1 (think that's right!) I take it that this means the infinite product therefore converges to 1?
euclid3.14 said:Have another querry on infinite products. what does [itex]\prod_{n=1}^{n}\left(1-1/(n+1)^2\right)[/itex]? converge to? I've write out the seris to get ((n+1)^2-1(n+2)^-1...) / ((n+1)^2*(n+2)^2...)
As the sum of 1/(n+1)2 converges I know that the infinite product must converge! :grumpy:
Doesn't it tend to one as n tends to infinity as you take the module of the series?shmoe said:Not so fast, you can only conclude that if it converges then it converges to 1. You can't sum it 2 terms at a time unless you know the thing is absolutely convergent. Consider [itex]\sum(-1)^n[/itex]. If you look at the partial sum of an even number of terms, you get Don't you mean 1 here? , but you wouldn't say this sum converges based on that.
You'll want to look at the product of an odd number of terms. What is [itex]\prod_{n=1}^{2k+1}\left(1+a_n\right)[/itex] in terms of k? Use the fact that you know what the product of an even number of terms is...
could simplify it to [itex]\prod_{n=1}^{k}\left((n+1)^2 -1)/(n+1)^2\right[/itex] or [itex]\prod_{n=1}^{k}\left(n^2 +2n /n^2 +2n +1\right)[/itex]shmoe said:Can you find a nice expression for [itex]\prod_{n=1}^{k}\left(1-1/(n+1)^2\right)[/itex] for any value of k? Try finding the product for k=1,2,3,4,5 and see if you can guess a nice formula (find more than 5 if you need to). Try to prove your formula is correct and use it to find the limit.
k 1 2 3 4 5
3/4 2/3 5/8 3/5
euclid3.14 said:Doesn't it tend to one as n tends to infinity as you take the module of the series?
There's a chance that I'm confused!
euclid3.14 said:could simplify it to [itex]\prod_{n=1}^{k}\left((n+1)^2 -1)/(n+1)^2\right[/itex] or [itex]\prod_{n=1}^{k}\left(n^2 +2n /n^2 +2n +1\right)[/itex]
Seems to be tending to 1/2 but how would could i explain that properly?Code:k 1 2 3 4 5 3/4 2/3 5/8 3/5
Cheers you the responce, it's been very helpful!
An infinite product is a mathematical expression that involves an infinite number of terms being multiplied together. It is similar to an infinite series, but instead of adding terms, we are multiplying them.
Infinite products are used in mathematics to represent certain functions or to approximate values of other mathematical expressions. They also have important applications in fields such as number theory and complex analysis.
Yes, an infinite product can converge to a finite value. This happens when the terms in the product become smaller and smaller as the number of terms increases. If the product of the terms approaches a finite value, then the infinite product itself will converge to that value.
To determine if an infinite product converges or diverges, we can use the ratio test or the root test. These tests compare the terms in the product to a geometric series or a p-series, and if the limit of the ratio or root is less than 1, then the product converges. If the limit is greater than 1, the product diverges.
The infinite product can be written as (1+a)(1+a2)(1+a4)(1+a6)... where each term is a power of a. This can then be simplified using the formula for the sum of a geometric series, giving us (1-a2)/(1-a), where a2n-1 represents the common ratio.