What is the sum of the series

arrostetzwyl

arrostetzwyl

Answered question

2022-03-14

What is the sum of the series k=112k1

Answer & Explanation

Alexa Lewis

Alexa Lewis

Beginner2022-03-15Added 5 answers

Ignoring the negative sign, you're looking for
1211+1221+1231+
but we have
12jk1=12jk+122jk+123jk+
If we rewrite the first expression using the second one, then your sum is
(121+122+123+)+(122+124+126+)+(123+126+129+)+
and 12r appears τ(r) times, where τ(r) is the number of divisors of r. Therefore your sum is
r1τ(r)2r
and this should allow you to compute it to any desired level of numerical accuracy fairly quickly. For example,
r=140τ(r)2r=2760281253717179869184=1.606695152
and the error here is less than r41r2r=84241<4×1011 since τ(r)<r for all r3
Of course this method works when 2 is replaced by any constant c>1

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