- #1

chwala

Gold Member

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- TL;DR Summary
- Showing that for any given ##ε>0## (no matter how small), there exists a number

##N##( depending on ##ε##) s.t ##|U_n - l| <ε##

I will create my own example on this- Phew atleast this concepts are becoming clearer ; your indulgence is welcome.

Let me have a sequence given as,

##Un = \dfrac {7n-1}{9n+2}##

##Lim_{n→∞} \left[\dfrac {7n-1}{9n+2}\right] = \dfrac {7}{9} ##

Now,

##\left[ \dfrac {7n-1}{9n+2} - \dfrac {7}{9} \right] = \left[ \dfrac {-23}{9(9n+2)} \right]##

when,

##\left[ \dfrac {23}{9(9n+2)} \right] <ε##

or

##\left[ \dfrac {9(9n+2)}{23} \right] >\left[\dfrac{1}{ε}\right]##

##9n+2 >\dfrac{23}{9ε}##

##9n > \dfrac{23}{9ε} -2##

##n > \dfrac{1}{9} \left[ \dfrac{23}{9ε} -2 \right]##

choosing ##N= \dfrac{1}{9} \left[ \dfrac{23}{9ε} -2 \right]## where say for e.g Let ##ε = 0.01##

##N= \dfrac{1}{9} [ 255.55555-2]=253.5555##

This means that, all terms beyond ##253## differ from ##\dfrac {7}{9} ## by an absolute value less than ##0.01##.

Let us check,

If ##N=300, u_n =\left|\dfrac{2099}{2702} - \dfrac {7}{9}\right|=|0.77683-0.7777|=0.00087<0.01##

Implying that if i pick a sequence less than ##253## then the proof will not hold. This is the time that i am getting to understand some analysis ...particularly of this epsilon thing!

Cheers. Any insight welcome.

Maybe the question that i may need to ask is how small can ##ε## be and how big can it be? I could say less than ##1## in order for the limit to exist.

Let me have a sequence given as,

##Un = \dfrac {7n-1}{9n+2}##

##Lim_{n→∞} \left[\dfrac {7n-1}{9n+2}\right] = \dfrac {7}{9} ##

Now,

##\left[ \dfrac {7n-1}{9n+2} - \dfrac {7}{9} \right] = \left[ \dfrac {-23}{9(9n+2)} \right]##

when,

##\left[ \dfrac {23}{9(9n+2)} \right] <ε##

or

##\left[ \dfrac {9(9n+2)}{23} \right] >\left[\dfrac{1}{ε}\right]##

##9n+2 >\dfrac{23}{9ε}##

##9n > \dfrac{23}{9ε} -2##

##n > \dfrac{1}{9} \left[ \dfrac{23}{9ε} -2 \right]##

choosing ##N= \dfrac{1}{9} \left[ \dfrac{23}{9ε} -2 \right]## where say for e.g Let ##ε = 0.01##

##N= \dfrac{1}{9} [ 255.55555-2]=253.5555##

This means that, all terms beyond ##253## differ from ##\dfrac {7}{9} ## by an absolute value less than ##0.01##.

Let us check,

If ##N=300, u_n =\left|\dfrac{2099}{2702} - \dfrac {7}{9}\right|=|0.77683-0.7777|=0.00087<0.01##

Implying that if i pick a sequence less than ##253## then the proof will not hold. This is the time that i am getting to understand some analysis ...particularly of this epsilon thing!

Cheers. Any insight welcome.

Maybe the question that i may need to ask is how small can ##ε## be and how big can it be? I could say less than ##1## in order for the limit to exist.

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