sequence limits

[forever119]

let [MATH](a_n)_{n=1}^\infty [/MATH] and [MATH](b_n)_{n=1}^\infty[/MATH] be convergent sequences and let their limits be [MATH]s[/MATH] and [MATH]t[/MATH] respectively. Let the sequence [MATH](c_n)_{n=1}^\infty[/MATH] defined by [MATH]c_n=\frac{1}{n}*\sum\limits_{k=1}^n a_k*b_{n-k+1} [/MATH] then find [MATH]lim_{n \to +\infty} c_n [/MATH] I know that in this case [MATH] lim_{n \to +\infty}\frac{1}{n}*\sum\limits_{k=1}^n a_k*b_k=s*t[/MATH] I tried to use this fact but couldnt proceed

 

[Deleted member 4993]

let [MATH](a_n)_{n=1}^\infty [/MATH] and [MATH](b_n)_{n=1}^\infty[/MATH] be convergent sequences and let their limits be [MATH]s[/MATH] and [MATH]t[/MATH] respectively. Let the sequence [MATH](c_n)_{n=1}^\infty[/MATH] defined by [MATH]c_n=\frac{1}{n}*\sum\limits_{k=1}^n a_k*b_{n-k+1} [/MATH] then find [MATH]lim_{n \to +\infty} c_n [/MATH] I know that in this case [MATH] lim_{n \to +\infty}\frac{1}{n}*\sum\limits_{k=1}^n a_k*b_k=s*t[/MATH] I tried to use this fact but couldnt proceed

Please show us your work till that point.

Please show us what you have tried and exactly where you are stuck.

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[Romsek]

I think the idea is that if you go far out enough in the sequences the vast bulk of the elements will be very close to their limit.

You are taking a vector of the first n elements of a and dotting this with the reverse of the first n elements of b.

If n is large the bulk of those elements will be very close to their limits and in the limit are you multiplying the limit of a and the limit of b n times and summing. Normalizing by 1/n gets you back to the product of the limits.

You'll need to do some sort of epsilon delta proof where given sum epsilon you have to figure out how big n needs to be to get [MATH] \left |c_n - s t \right|<\epsilon[/MATH]

 

[Steven G]

[math]lim_{n \to +\infty}\frac{1}{n}*\sum\limits_{k=1}^n a_k*b_k=s*t[/math]I can't buy that line above for a second.
Sure without the 1/n in the limit, the limit is st but with the 1/n the lim will be 0.

Remember that the limit of a product is the product of the limits which is 0*st=0

 

[Romsek]

let [MATH]a_n = A, b_k = B[/MATH]
[MATH]\sum \limits_{k=1}^n a_k b_k = n A B\\ \dfrac 1 n \sum \limits_{k=1}^n a_k b_k = AB[/MATH]
So I'm not sure what you're thinking when you say the [MATH]\dfrac 1 n[/MATH] in front causes everything to go to zero.

 

[Steven G]

let [MATH]a_n = A, b_k = B[/MATH]
[MATH]\sum \limits_{k=1}^n a_k b_k = n A B\\ \dfrac 1 n \sum \limits_{k=1}^n a_k b_k = AB[/MATH]
So I'm not sure what you're thinking when you say the [MATH]\dfrac 1 n[/MATH] in front causes everything to go to zero.

I agree with everything you wrote. I just do not know why you wrote it. The OP said let [math](a_n)_{n=1}^\infty\ and\ (b_n)_{n=1}^\infty[/math] be convergent sequences and let their limits be s and t respectively. The OP never said that that the terms of a_n and b_n were constants. Does that change everything?

 

[Steven G]

I am beginning to see that I am wrong. I just need to see why the OP is correct. I just need to read all of this again. Thanks!

 

[Romsek]

I agree with everything you wrote. I just do not know why you wrote it. The OP said let [math](a_n)_{n=1}^\infty\ and\ (b_n)_{n=1}^\infty[/math] be convergent sequences and let their limits be s and t respectively. The OP never said that that the terms of a_n and b_n were constants. Does that change everything?

That was just a simple example of how taking the average of something doesn't necessarily make it go to zero.

 

[forever119]

[math]lim_{n \to +\infty}\frac{1}{n}*\sum\limits_{k=1}^n a_k*b_k=s*t[/math]I can't buy that line above for a second.
Sure without the 1/n in the limit, the limit is st but with the 1/n the lim will be 0.

Remember that the limit of a product is the product of the limits which is 0*st=0

have you ever read any real analysis book???

 

[Steven G]

have you ever read any real analysis book???

A few of them.