Let's have a sequence [tex]x_n=\sum_{k=1}^{k=n}{\frac{1}{2^{\sqrt{k}}}[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

We must prove it is convergent.

First thought, let's try to prove it is monotonic and bounded, which means convergence of sequence.

Monotonicity is easy, [tex]\forall n \in N: x_{n+1}-x_n = \frac{1}{2^{\sqrt{n+1}}} > 0[/tex]

So, sequence is increasing. Next, I should prove it has an upper bound, but I am not able to come up with bigger sequence which would have a positive finite limit.

Any ideas ?

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# Proof of sequence convergence

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