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Question Number 132230 by pticantor last updated on 12/Feb/21

    S=Σ_(k=1) ^(+∞) ((1/( (√(k^2 −1))))−(1/( (√(k^2 +1)))))  S =l or S=∞  ???

$$ \\ $$$$ \\ $$$$\boldsymbol{{S}}=\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{+\infty} {\sum}}\left(\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{k}}^{\mathrm{2}} −\mathrm{1}}}−\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{k}}^{\mathrm{2}} +\mathrm{1}}}\right) \\ $$$$\boldsymbol{{S}}\:={l}\:{or}\:\boldsymbol{{S}}=\infty\:\:??? \\ $$

Answered by JDamian last updated on 12/Feb/21

∞ because (1/( (√(k^2 −1))))=∞     k=1

$$\infty\:{because}\:\frac{\mathrm{1}}{\:\sqrt{{k}^{\mathrm{2}} −\mathrm{1}}}=\infty\:\:\:\:\:{k}=\mathrm{1} \\ $$

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