Let 1 2 3 0 1 where is a constant such that 1 0 Then we see that 1 1 1 1 1 1 0 But 1 0 Hence 0 for all values of That is to say 1 2 3 4 1 1 2 2 3 1 But it is known that 2 Putting 1 in 1 and using 2 we obtain 1 1 2 2 3 3 4 3 1 2 3 4 3 2 ID: 23126 Download Pdf

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Let 1 2 3 0 1 where is a constant such that 1 0 Then we see that 1 1 1 1 1 1 0 But 1 0 Hence 0 for all values of That is to say 1 2 3 4 1 1 2 2 3 1 But it is known that 2 Putting 1 in 1 and using 2 we obtain 1 1 2 2 3 3 4 3 1 2 3 4 3 2

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On the sum of the square roots of the ﬁrst natural numbers Journal of the Indian Mathematical Society, VII, 1915, 173 – 175 1. Let ) = 1+ 2+ 3+ =0 )+ +1) where is a constant such that (1) = 0 Then we see that +1) = +1)+ +1) +1)+ +1) )+ +1) = 0 But (1) = 0 Hence ) = 0 for all values of . That is to say 1+ 2+ 3+ 4+ +1) +1)+ +2) +2)+ +3) (1) But it is known that (2) Putting = 1 in (1) and using (2), we obtain 1) 1+ 2) 2+ 3) 3+ 4) = 3 1) 2) 3) 4) (3) 2. Again let ) = 1 1+2 ... 40 =0 )+ +1)]

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60 Paper 9 where is a constant such that (1) = 0 Then we have +1) =

+1) +1) +1) +1)+ +1) +1)+ +1) −{ 40 +1) = 0 But (1) = 0 Hence ) = 0 In other words 1+2 2+3 3+ 40 +1) +1)+ +2) +2)+ +3) (4) But it is known that 16 (5) It is easy to see from (4) and (5) that 1) 1+ 2) 2+ 3) 3+ 4) = 15 1) 2) 3) 4) (6) 3. The corresponding results for higher powers are not so neat a s the previous ones. Thus for example 1+2 2+3 3+ 24 96 +1) +1)+ +2) 224 +1) +1)+ +2) +2)+ +3) ; (7) 1+2 2+ 24 384 192 +1) +1)+ +2) 1152 +1) +1)+ +2) ]; (8) and so on.

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On the sum of the square roots of the ﬁrst natural numbers 61 The constants ,C ,... can be ascertained

from the well-known result that the constant in the approximate summation of the series +2 +3 is 2Γ( (2 cos πr, (9) provided that the real part of is greater than 1. 4. Similarly we can shew, by induction, that +2 +1) +1) +1)+ +2) +1)( +2) (10) The value of can be determined as follows: from (10) we have (2 (2 (11) as . Also (2 (2 (12) as Now subtracting (12) from (11) we see that (2 (1 2) as That is to say (1+ 2) (13) We can also shew, by induction, that 1+ 2+ 3+ 24 24 +1) +1) +1)+ +2) +1)( +2) (14) The asymptotic expansion of 1+ 2+ 3+ for large values of can be shewn to be 24 1920

9216 (15) by using the Euler-Maclaurin sum formula.

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