# proving the convergence of a sequence..

by transgalactic
Tags: convergence, proving, sequence
 P: 1,399 $$2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}$$ 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) $$a_1=2$$ $$a_2=2.5$$ so i guess its increasing suppose n=k is true: $$a_{k-1}a_{k-1}\\$$ $$\frac{1}{a_k}<\frac{1}{a_{k-1}}\\$$ $$2+\frac{1}{a_k}<2+\frac{1}{a_{k-1}}\\$$ [tex] a_{k+1}

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