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Real analysis

Using the definition of convergence of a sequence
show that the following sequences converge to the proposed limit: 1-lim 1/(6n^2+1)=0
2-lim 2/sqrt[n+3]=0
3-lim (3n+1)/(2n+5)=3/2

Solution Preview

1-
Let epsilon> 0 and let N be a natural number such that 1/(6N^2)< epsilon. Then if n>= N we will have:
|1/(6n^2+1)|< |1/(6n^2)|<= |1/(6N^2)|< epsilon
And that is what we were ...

Solution Summary

This uses the definition of convergence of a sequence to show that sequences converge to the proposed limits.

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