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[tex]2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}[/tex]

etc..

(the sequence consists only from positive number so the sum is not negative)

in order to prove that its convergent i need to prove monotonicity and boundedness

monotonicity:(by induction)

[tex]a_1=2[/tex]

[tex]a_2=2.5[/tex]

so i guess its increasing

suppose n=k is true:

[tex]a_{k-1}<a_k[/tex]

prove n=k+1 ([tex]a_{k}<a_{k+1}[/tex])

[tex]

a_k>a_{k-1}\\

[/tex]

[tex]

\frac{1}{a_k}<\frac{1}{a_{k-1}}\\

[/tex]

[tex]

2+\frac{1}{a_k}<2+\frac{1}{a_{k-1}}\\

[/tex]

[tex]

a_{k+1}<a_k

[/tex]

i proved the opposite :)

so this is weird.

the answer in the book tells me to split the sequence into odd /even sub sequences

the one is ascending and the other its descending.

i cant see how many sub sequences i need to split it to

maybe its 5 or 10

what is the general way of solving it.

and how you explained that i proved the opposite

etc..

(the sequence consists only from positive number so the sum is not negative)

in order to prove that its convergent i need to prove monotonicity and boundedness

monotonicity:(by induction)

[tex]a_1=2[/tex]

[tex]a_2=2.5[/tex]

so i guess its increasing

suppose n=k is true:

[tex]a_{k-1}<a_k[/tex]

prove n=k+1 ([tex]a_{k}<a_{k+1}[/tex])

[tex]

a_k>a_{k-1}\\

[/tex]

[tex]

\frac{1}{a_k}<\frac{1}{a_{k-1}}\\

[/tex]

[tex]

2+\frac{1}{a_k}<2+\frac{1}{a_{k-1}}\\

[/tex]

[tex]

a_{k+1}<a_k

[/tex]

i proved the opposite :)

so this is weird.

the answer in the book tells me to split the sequence into odd /even sub sequences

the one is ascending and the other its descending.

i cant see how many sub sequences i need to split it to

maybe its 5 or 10

what is the general way of solving it.

and how you explained that i proved the opposite

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