Mr Infinity

Algebra Level 3

Mr. Infinity made a stadium with infinite number of seats. Where all the seats are numbered as 1 , 2 , 3.... 1,2,3.... . For a special guest stadium committee take the decision to transfer the spectators from n t h n^{th} numbered seat to ( n + 1 ) t h (n+1)^{th} numbered seat. For this they refund him { 1 n − 1 n + 1 \frac{1}{n}-\frac{1}{n+1} } dollar . How many dollars stadium committee needs to transfer seat by this process?

NOTE: If the answer is infinity then give your answer -0.5


The answer is 1.00.

Farhabi Mojib by Farhabi Mojib , 23, Bangladesh

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2 solutions

There has to be paid 1-1/2 + 1/2 - 1/3 +1/3 - 1/4 +1/4 - 1/5 and so on.... so if there were n spectators there had to be paid 1 - 1/n. Since no approaches infinity 1/n approaches 0. Hence there has to be paid 1.

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Farhabi Mojib
Nov 25, 2016

1 n − 1 n + 1 \frac{1}{n} - \frac{1}{n+1} = 1 n ( n + 1 ) \frac{1}{n(n+1)}

We know,
∑ k = 1 n 1 k ( k + 1 ) = n n + 1 \sum_{k=1}^n \frac{1}{k(k+1)} = \frac{n}{n+1}
and as n approaches infinity n n + 1 \frac{n}{n+1} approaches to 1.

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