How do you find the 4-th partial sum of the infinite series $sum_(n=1)^oo(1/sqrt(n)-1/sqrt(n+1))$ ?

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Answer 1 · 2014-09-20T19:44:36

$S_4=(1/sqrt{1}-1/sqrt{2})+(1/sqrt{2}-1/sqrt{3})+(1/sqrt{3}-1/sqrt{4})+(1/sqrt{4}-1/sqrt{5})$

by regrouping,

$=1+(-1/sqrt{2}+1/sqrt{2})+(-1/sqrt{3}+1/sqrt{3})+(-1/sqrt{4}+1/sqrt{4})-1/sqrt{5}$

by cancelling intermediate terms,

$=1-1/sqrt{5}$

How do you find the 4-th partial sum of the infinite series sum_(n=1)^oo(1/sqrt(n)-1/sqrt(n+1))sum_(n=1)^oo(1/sqrt(n)-1/sqrt(n+1)) ?